.^v 


y<l> 


X#^ 


LIBRAR.^,n.j. 

OF  THE 

University  of  California. 


OIF^T  OK 


zAccession  Class  ^ 


AMERICAN  B-OK  ^'^0, 

A.  I^.  GUNN,  G-eii']  .g'-, 

204   PINE   STR_;^T, 

SAN   FRANCISCO. 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/elementsofarithmOOmilnrich 


ELEMENTS  OF  ARITHMETIC 


FOE    PKIMARY    AND    INTERMEDIATE    CLASSES    IN 
PUBLIC  AND  PRIVATE  SCHOOLS 


BY 

WILLIAM   J.   MILNE,   Ph.D.,  LL.D. 

President  of  New  York  State  Normal  College,  Albany,  N  Y. 


or  xMx 

UNIVERSITY 


NEW   YORK  .;.  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


MILNE'S  MATHEMATICS 


Milne's  Elements  of  Arithmetic 

Milne's  Standard  Arithmetic 

Milne's  Mental  Arithmetic 

Milne's  Elements  of  Algebra 

Milne's  Grammar  School  Algebra 

Milne's  High  School  Algebra 
Milne's  Plane  and  Solid  Geometry 
Milne's  Plane  Geometry — Separate 


Copyright,  1893,  by  American  Book  Company 

Elmts.  Ar. 

E-P68 


PREFACE. 


THIS  work  has  been  prepared  to  meet  a  demand  for  a  book 
that  contains,  within  brief  compass,  an  elementary  course  in 
arithmetic.  It  is  also  designed  to  be  an  introduction  to  the  more 
thorough  and  exhaustive  treatment  of  the  science  which  is  given  in 
the  author's  Standard  Arithmetic. 

The  first  part  of  the  book  contains  exercises  arranged  in  such  a 
manner  that  they  secure  an  easy  and  natural  development  of  the 
ideas  of  numbers,  and  they  afford  sufficient  practice  to  fix  the 
results  in  memory.  Ready  knowledge  of  results  can  be  secured 
only  by  frequent  repetition  and  thorough  drill,  and  the  lessons  that 
have  been  prepared  are  believed  to  be  admirably  adapted  to  produce 
expertness  in  computing,  without  sacrificing  interest  in  the  subject 
through  constant  practice  with  abstract  numbers. 

When  the  more  systematic  treatment  of  the  science  is  presented, 
the  pupil  is  led  by  natural,  progressive,  and  logical  steps  to  an 
understanding  of  the  definitions,  principles,  processes,  and  rules, 
before  he  is  required  to  state  them ;  consequently,  all  definitions, 
principles,  and  rules  are  but  the  expressions  of  what  he  already 
knows.  It  is  evident,  therefore,  that  the  plan  pursued  in  the  work 
will  develop  in  the  student  the  habit  of  investigating  for  himself 
any  subject  which  may  claim  his  attention,  and  this  is  an  extremely 
important  part  of  proper  teaching. 

The  number  of  oral  examples  is  large  enough  to  supply  all  the 
necessary  training  in  "  mental "  arithmetic,  and  the  frequent  exer- 
cises for  review  will  fix  in  the  memory  all  the  knowledge  of  the 
science  that  has  been  acquired.  The  treatment  of  the  subjects  has 
been  adapted  to  the  comprehension  of  young  pupils,  and  subjects 
which  are  too  difficult  for  them  have  been  omitted  ;  and  yet  in  no 
instance  has  accuracy  of  statement  or  correctness  of  process  been 
sacrificed  in  what  has  been  presented. 

85i2o  3 


4  PREFACE. 

Though  the  treatment  of  the  subjects  is  brief  and  elementary,  a 
student  who  goes  no  farther  in  his  arithmetical  studies  than  this 
work  will  take  him,  will  have  a  knowledge  of  the  science,  sufficient 
to  enable  him  to  perform,  with  accuracy  and  intelligence,  most  of 
the  processes  required  in  ordinary  business  life. 

The  author  desires  to  express  his  indebtedness  to  George  I. 
Aldrich,  A.M.,  Superintendent  of  Schools  of  Newtonville,  Mass.,  for 
valuable  aid  in  the  preparation  of  the  earlier  parts  of  the  book,  and 
for  important  suggestions  upon  the  treatment  of  subjects  through- 
out the  work.  His  reputation  as  an  originator  of  rational  and 
scholarly  methods  of  teaching  arithmetic  is  a  guarantee  of  the 
value  of  his  services. 

"The  Elements  of  Arithmetic  "  is  submitted  to  the  public  with 
the  hope  that  it  may  prove  to  merit  general  favor  on  account  of  the 
natural  and  rational  development  of  the  subject,  and  because  of 
its  peculiar  adaptation  to  the  class  of  students  for  whose  use  it 
was  written. 


WILLIAM  J.   MILNE. 


State  Normal  College, 
Albany,  N.Y. 


SUGGESTIONS   TO   TEACHERS. 


1.  It  must  be  borne  iu  mind  that  the  pupils  should  first 
be  made  familiar  with  the  combinations  of  numbers^  before 
any  explanations  or  analyses  are  required. 

2.  The  first  part  of  this  book  is  designed  to  give  pupils 
the  knowledge  of  numbers;  and  it  is  suggested,  therefore^ 
that  the  teacher  ask  for  results  only. 

3.  If  explanations  of  processes  or  analyses  of  problems 
are  required,  they  will  divert  the  attention  of  the  pupils 
from  the  results,  and  thus  hinder  their  progress. 

4.  The  lessons  which  are  given  are  designed  to  be  typical 
rather  than  complete,  and  the  thoughtful  teacher  will  devise 
additional  problems  of  a  character  similar  to  those  given  in 
the  book. 

5.  By  going  over  the  same  lessons  frequently  the  pupil 
will  become  familiar  with  numbers ;  but  there  is  only  one 
way  in  which  thorough  knowledge  of  numbers  can  be  ac- 
quired, and  that  is  by  repetition  of  the  combinations  within 
brief  intervals  of  time. 

6.  The  examples  should  at  first  be  given  in  connection 
with  objects  which  can  be  conveniently  obtained.  From 
these  the  transition  is  easy  to  concrete  examples,  in  which 
the  splints,  or  counters,  or  buttons,  or  beans,  stand  for  other 

5 


6     ,  SUGGESTIONS   TO   TEACHERS. 

units  ;  but  too  early  drill  upon  abstract  numbers  is  unfavor- 
able to  the  develoioment  of  a  proper  comprehension  of 
arithmetic,  and  makes  the  pupils  mechanical  rather  than 
thoughtful. 

7.  It  will  be  found  advantageous  to  encourage  the  pupils 
to  form  examples  for  themselves.  Such  a  plan  gives  a 
pleasing  variety  to  the  work  of  the  class,  and  makes  the 
pupils  thoughtful  regarding  the  work.  Judicious  use  of 
this  suggestion  will  aid  in  inspiring  interest  in  the  work. 

8.  The  work  upon  the  slate,  or  written  work,  should 
come  only  after  the  oral  work,  and  it  should  be  given  only 
for  purposes  of  review  after  the  numbers  have  been  thor- 
oughly understood. 

9.  Care  should  be  taken  with  the  written  work,  that  the 
children  make  the  figures  in  accordance  with  proper  models, 
and  that  everything  is  done  neatly. 

10.  Be  sure  to  require  the  students  to  answer  all  the 
development  exercises  which  precede  every  subject.  These 
exercises  will  cause  the  pupils  to  apprehend  the  principles 
that  are  employed  in  the  solution  of  the  problems  that 
follow,  and  they  will  discourage  the  tendency  on  the  part 
of  the  children  to  give  mechanical  or  formal  solutions. 

11.  Do  not  fail  with  every  lesson  to  review  the  work  of 
previous  days,  for  it  is  only  in  this  way  that  pupils  become 
familiar  with  results  and  processes,  and  success  in  arith- 
metical processes  is  dependent  to  a  great  degree  upon  con- 
stant drill. 


CONTENTS. 


PAGE 

Exercises  on  Numbers  1  to  100 9 

Notation  and  Numeration o     .     .     .  78 

Notation  and  Numeration  of  United  States  Money  ....  85 

Roman  Notation 87 

Addition 89 

Subtraction 95 

Review 99 

Multiplication 102 

Review 114 

Division 119 

Review 130 

Factoring 138 

Tests  of  Divisibility 140 

Cancellation .  141 

Fractions o     .     . 145 

Reduction     o »     .  151 

Addition  .     , ,     ......  156 

Subtraction 159 

Multiplication    « 162 

Division    ........ 109 

Aliquot  Parts 179 

Review 182 

7 


8 


CONTENTS. 


Decimal  Fractions     o 

Reduction  o  »  , 
Addition  .... 
Subtraction  .  .  . 
Multiplication  .  . 
Division  .... 
Accounts  and  Bills 
Review  .... 
Denominate  Numbers 
Measures  of  Length 
Measures  of  Surface 
Measures  of  Volume 
Liquid  Measures  . 
Dry  Measures  .  . 
Avoirdupois  Weight 
Troy  Weight  .  . 
Apothecaries'  Weight 
Divisions  of  Time  , 
Counting  .... 
Stationers'  Table  . 
Reduction  .  .  , 
Addition  .  ,  .  . 
Subtraction  .  .  . 
Multiplication  .  . 
Division  .... 
Practical  Measurements  . 
Interest  .  .  .  o  .  .  , 
General  Review   ,00. 


PAGE 

188 
191 
193 
194 
195 
198 
201 
203 
205 
205 
206 
207 
208 
209 
209 
210 
210 
211 
213 
213 
214 
217 
218 
219 
220 
221 
226 
229 


u»  ^  fiA 


UNIVERSITY 

ELEMENTS  OE^  A^RITHMETIC. 


3i»4C 


1.  Hold  up  your  right  hand. 

Hold  up  your  left  hand. 

How  many  hands  have  you  ? 

Place  your  right  hand  behind  you. 
How  many  hands  can  I  see  now  ? 
Place  your  left  hand  behind  you. 
How  many  hands  are  behind  you? 
How  many  feet  have  you  ? 
How  many  eyes  have  you  ? 
How  many  heads  have  you  ? 
Name  one  thing  you  can  see  on  my  desk. 
Show  me  two  blocks. 
Give  Annie  one  block. 
Clap  your  hands  once. 
Say  hall  twice. 
Make  a  mark  on  your  slate. 

9 


10  ELEMENTS  OF  ARITHMETIC. 

Make  another  mark. 

How  many  marks  have  you  made? 

One  mark  and  one  mark  are  how  many  marks  ? 

Kub  out  one  mark.     How  many  marks  are  left? 

This  figure  stands  for  one  —  /. 

Tliis  figure  stands  for  two  —  2- 

Say  one  word. 

Say  two  words. 

Point  to  one  book. 

Point  to  one  school-desk. 

Hold  up  two  fingers. 

Tell  me  two  boys'  names. 

Tell  me  a  girl's  name. 

How  many  feet  has  a  boy  ? 

How  many  noses  has  he  ? 

How  many  ears  has  he  ? 

Make  as  many  marks  as  you  have  hands. 

Make  the  figure  that  stands  for  two  things. 

Make  as  many  marks  as  you  have  feet. 

Make  the  figure  that  stands  for  one  thing. 

If  I  have  two  cents  and  give  away  one  of  them,  how 
many  shall  I  have  left  ? 

If  you  should  liave  one  apple,  and  a  girl  should  give 
you  one  more,  how  many  apples  would  you  have  ? 

If  you  did  not  go  to  school  on  Monday  and  on  Friday, 
how  many  days  were  you  absent  that  week  ? 
-    Make  pictures  of  two  squares.     Q  []] 

Make  pictures  of  two  circles.      Q  O 

Make  pictures  of  two  triangles,      j^  ^ 

n  and  Q  are  how  many  squares? 

O  i^^^^d  O  ^^^^  5^0 w  many  circles  ? 

A  and  /\  are  how  many  triangles? 


THREE  —  3. 


11 


2.  How  many  boys  are  picking 
up  nuts  ? 

How  many  boys  are  in  the  tree? 
How  many  boys  do  you  see  in 
the  picture?     If  one  of  the  boys 
should  go  home,  how  many  would  be  left  ? 
Show  me  three  pencils. 
Make  three  dots  on  your  slate. 
Rub  out  one  dot.     How  many  are  left? 
Rub  out  another.     How  many  have  you  rubbed  out  ? 
How  many  are  left? 

Now  put  enough  dots  with  this  one  dot  to  make  three 
dots. 

One  dot  and  two  dots  are  how  many  dots? 
James   gave  me  two  marbles,  and  Arthur  gave   me 
one.     How  many  marbles  did  they  give  me? 

Two   marbles   and  one  marble   are   how   many  mar- 
bles? 

If  I  lost  one  of  them,  how  many  had  I  left? 
Three  marbles  less  one  marble  are  how  many  marbles? 
I  see  one   bird  on  the  ground,  one  in  a  bush,  and 
one  flying.     How  many  birds  do  I  see  ? 


12 


ELEMENTS  OF   ARITHMETIC. 


This  figure  stands  for  three  —  J. 

Write  the  figures  that  stand  for  one,  two,  and  tliree. 

Tell  me  what  tliis  shows :  •  •  and  •  are  •  •  •. 

|~|  and  [31  n  are  how  many  squares? 

/\^  and  A  A  ^^'®  ^^w  many  triangles? 


3.  How  many  boys  do  you  see  in  the  picture  ? 

How  many  girls  do  you  see  ? 

How  many  children  are  there  ? 

Three  boys  and  one  girl  are  how  many  children  ? 

Take  four  blocks.  Play  that  they  are  little  birds. 
Four  little  birds  were  picking  up  crumbs.  When  one 
flew  away,  how  many  were  left?  Soon  afterward,  two 
more  flew  away.  How  many  had  then  flown  away? 
How  many  were  left  ? 

How  many  books  has  the  largest  boy  in  the  pic- 
ture ? 

How  many  has  the  next  largest? 

If  the  largest  boy  should  carry  all  the  books,  how 
many  would  he  carry  ? 


FOUR  — 4.  13 

Susie  and  her  sister  found  a  nest  with  four  eggs  in  it. 
Her  sister  took  two  of  them.  How  many  were  left  for 
Susie  ? 

Four  eggs  less  two  eggs  are  how  many  eggs  ? 

In  four  eggs,  how  many  twos  are  there  ? 

How  many  sides  has  a  square  ?     □ 

How  many  sides  has  a  triangle  ?     /\ 

How  many  triangles  can  you  form  with  four  sticks  ? 
Try  -It. 

At  a  cent  apiece,  how  many  pears  can  you  buy  with, 
four  cents  ? 

This  figure  stands  for  four  —  -^. 

Write  the  figures  / ,  2,  3,  ^  on  your  slate. 

How  many  one-cent  stamps  can  you  buy  for  four  cents  ? 

When  pencils  cost  two  cents  each,  what  will  two 
pencils  cost? 

James  was  two  years  old,  and  his  sister  Grace  was 
two  years  older.     How  old  was  Grace  ? 

Make  two  squares  and  write  the  word  two  beside 
them.     OO,   two. 

Make  three  circles  and  write  the  word  three  beside 
them.     OOO?   three. 

Make  four  triangles  and  write  the  word  four  beside 
them.     AAAA.   four. 

How  many  squares  are  D  D  and  D  D  ? 

Here  is  a  shorter  way  of  writing  "  Two  and  two  are 
four":  2  +  2  =  4. 

The  sign  +  means  and^  and  the  sign  =  means  are. 

Read  2  +  1  =  3;  1  +  1  =  2;  3  +  1  =  4;  1  +  3  =  4. 

Copy  and  complete  : 

1+3=?         2+2=?         1+1=?         1+2=? 


14  ELEMENTS   OF   ARITHMETIC. 

4,  How  many  fingers  have  you  on  one 
hand  ? 

If  you  do  not  count  the  thumb,  how 
many  are  there  ? 

Five  fingers  less  one  finger  are  how 
many  fingers  ? 

Point  to  five  children. 
Make  five  i's  on  your  slate. 
How   many  books   must  I  put  with 
four  books  to  make  five  books  ? 

How  many  joints  has  your  little  fin- 
ger? 

How  many  joints  has  your  thumb? 
How  many  joints  have  both  your   little    finger   and 
your  thumb  ? 

Three  things  and  two  things  are  how  many  things  ? 
Mary  has  five  pencils.    If  only  two  of  them  are  sharp, 
how  many  are  dull  ? 

Alice  had  five  yards  of  ribbon.  She  gave  Sarah  three 
yards  of  it.     How  many  yards  had  she  left  ? 

Place  five  splints  on  your  desk  in  this  way:  II  II  I. 
How  many  twos  are  there  in  five,  and  hoAv  many  are 
left  over  ? 

If  you  had  two  two-cent  pieces  and  a  cent,  how  much 
money  would  you  have  ? 

What  is  another  name  for  five  cents  ? 
If  you  had  a  half-dime    and   spent   one    cent   for   a 
pencil   and   four  cents  for  marbles,  how  much  money 
would  you  have  left  ? 

This  figure  stands  for  five  —  S'- 

Write  the  figures  /,  £>^  d,  ^,  6  on  your  slate. 

How   many    cents   are   three    cents  and  two  cents? 


FIVE  — 5. 


Four  cents  and  one  cent?  One  cent  and^ four  cents? 
Two  cents  and  three  cents  ? 

How  much  will  be  left  out  of  five  cents  if  I  buy  two 
two-cent  stamps  ? 

It  takes  five  days  to  go  to  California.  When  we  have 
traveled  three  days,  how  many  days  have  we  to  travel? 

I  paid  three  cents  for  a  paper  and  had  two  cents  left. 
How  much  money  had  I  at  first  ? 

Write  a  figure  that  means  as  much  as  three  and  two. 
Three  and  one.     Two  and  one.     One  and  one. 

How  many  are 


less  four  apples  ? 

Here  is  a  shorter  way  of  writing  "  Five  less  four  are 
one":  5-4  =  1. 

The  sign  —  means  less.     What  does  =  mean  ? 


Copy  and  complete 

: 

5-3=? 

3-2=? 

2  +  3=? 

5-5=? 

5-4=? 

4  +  1=? 

4-1=? 

5-1=? 

3  +  2=? 

4-2=? 

4-3=? 

2  +  2=? 

O    one 

1. 

O  O    two 

2. 

O   O   O    three 

3. 

O 

O   O   O    four 

4. 

o  o 

O  O   O    five 

5. 

16 


ELEMENTS   OF   ARITHMETIC. 


5.  Cut    an    apple    into 
two  equal  parts. 

Draw  a  square  and  di- 
vide it  into  two  equal  parts. 
How  have  I  divided  the 
circle  placed  below? 

Each   of  these  parts  is 
called  a  half. 

How   many   halves    are 
there  in  an  apple  ? 
How  many  halves  are  there  in  a  square  ? 
Take  four  blocks.     Place  them  in  two  equal  groups. 
How  many  are  there  in  each  group  ?     What  is  the  name 
of  each  group,  or  part  ? 

One  block  is  what  part  of  two  blocks  ? 
James  had  two  nuts.     He  gave  half  of  them  to  me. 
How  many  did  he  give  me  ? 

What  part  of  the  nuts  did  he  keep  ? 

Divide  four  oranges  equally  between  Mabel 
and  Annie.  What  part  of  the  four  oranges 
will  each  have  ? 

How  many  of  the  oranges  will  each  have  ? 
*  What  is  half  of  four  books  ? 
Two  cents  is  half  of  how  many  cents  ? 
Mary  picked  four  pints  of  berries,  but  spilled 
half  of  them.     How  many  did  she  spill  ? 
Henry  made  four  cents  by  selling  newspapers,  and 
gave  half  of  it  to  his  sister.     How  much  did  he  give  to 
her? 

The  wages  of  a  boy  were  four  dollars  per  week. 
How  much  did  he  earn  in  half  a  week  ?  How  much  in 
a  week  and  a  half  ? 


SIX— 6. 


17 


6.  How  many  bananas  do  you  see  in  the  bunch? 

How  many  bananas  are  there  in  the  dish? 

Five  bananas  and  one  banana  are  how  many? 

Show  me  six  fingers. 

Draw  a  picture  of  a  pen  with  six  rabbits  in  it. 

Give  me  one  splint;  now  give  me  five  more.  How 
many  have  I  now  ? 

Jamie  had  a  half-dime  and  a  cent.  To  how  many 
cents  was  that  equal  ? 

Henry  had  six  pears,  but  he  ate  one.  How  many  had 
he  left? 

There  were  six  rolls  on  the  plate  before  supper,  and 
only  one  after  supper.     How  many  had  been  eaten  ? 

Johnnie  is  four  years  old  to-day.  How  old  will  he  be 
in  two  more  years  ? 

Four  years  and  two  years  are  how  many  years  ? 

It  is  two  miles  from  Lizzie's  house  to  school,  and 
four  more  miles  to  the  station.  How  far  does  Lizzie 
live  from  the  station  ? 

EL.    OF   AR.  — 2 


18  ELEMENTS   OF   ARITHMETIC. 

I  must  write  six  letters.  When  I  have  written  four 
of  them,  how  many  more  have  I  to  write  ? 

Make  six  lines  on  your  slate.  Cross  out  two  of  them. 
How  many  are  not  crossed  out? 

How  many  arms,  hands,  and  feet  have  you  ? 

How  many  eyes,  ears,  and  cheeks  have  you  ? 

Two  cents  and  two  cents  and  two  cents  are  how 
many  cents  ? 

If  you  had  six  cents,  how  many  two-cent  stamps 
could  you  buy  with  the  money  ? 

How  much  must  I  pay  for  three  sponges,  if  each  one 
costs  me  two  cents  ? 

How  many  boots  are  there  in  three  pairs  of  boots  ? 

Henry  has  six  mittens.  How  many  pairs  of  mittens 
has  he  ? 

Bertie  found  three  eggs  in  one  nest  and  three  in 
another.     How  many  eggs  did  he  find  ? 

Two  threes  are  how  many  ? 

Take  six  blocks.  Give  me  half  of  them.  How  many 
did  you  give  me  ? 

Nellie  had  six  eggs,  but  used  half  of  them  to  make 
a  cake.  How  many  did  she  use?  What  part  of  the 
eggs  was  left?     How  many  eggs  were  left? 

In  a  ,  class  of  six  pupils,  half  of  the  children  are 
sitting  and  half  are  standing.     How  many  are  sitting  ? 

Willie  had  six  cents.  He  spent  it  for  candy  at  a 
cent  a  stick.     How  many  sticks  did  he  buy  ? 

If  he  had  bought  marbles  with  it  at  three  cents 
apiece,  how  many  would  he  have  bought? 

This  is  the  figure  that  stands  for  six  —  6- 

Write  the  figures  f ,  2,  d,  ^,  6,  6,  on  your  slate. 

What  is  the  cost  of  three  pencils,  at  two  cents  apiece  ? 


ONE   THIRD. 


19 


5  +  1=? 

2+?  =  6 
3  +  3=? 
?+2  =  6 


SLATE   WORK. 
6_3=?  ?+5 


6-D=? 

6-?  =  4 


6 

3+?  =  6 
6-1=? 
6-?  =  2 


8  +  2=? 
4-3=? 
5-4=? 
1  +  3=? 


7.  How  have  I  divided  this  circle  ? 

Draw  a  line  and  divide  it  into  three  equal 
parts. 

Cut  an  api^le  into 
three  equal  parts. 

Each  of  these  parts 
is  called  a  third. 

How  many  thirds  are 
there  in  a  circle  ?  In  a 
line  ?  In  an  apple  ?  In 
anything  ? 

Give  me  a  third  of  a  splint. 

Take  three  pebbles.  Divide  them  into  three  equal 
parts.     Give  me  a  third  of  three  pebbles. 

Now  take  six  pebbles.  Divide  them  into  three  equal 
parts.     Give  me  a  third  of  six  pebbles. 

What  is  a  third  of  six  horses  ? 

If  a  pie  costs  six  cents,  what  will  a  third  of  a  pie 
cost? 

Henry  had  six  apples  which  he  divided  equally  among 
three  friends.     How  many  did  each  receive  ? 

School  is  in  session  six  hours  per  day,  but  the  little 
children  are  not  required  to  attend  all  the  time.  If  they 
have  one  third  of  the  time  for  play,  how  many  hours  do 
they  play  ? 


20 


ELEMENTS  OF  ARITHMETIC. 


8.  Get  a  pint  measure  and  a  quart  measure. 

Which  is  the  pint  measure?  Which  is  the  quart 
measure  ? 

Fill  the  pint  measure  with  water.  Pour  it  into  the 
quart  measure.     Does  it  fill  it? 

Fill  the  pint  measure  again  and  empty  it  into  the 
quart  measure.  Is  the  quart  measure  full  now?  How 
many  pints  of  water  are  there  in  it  ? 

How  many  pints  of  water  does  it  take  to  make  a 
quart  of  water  ?  How  many  pints  of  milk  make  a  quart 
of  milk  ?  How  many  pints  of  berries  make  a  quart  of 
berries  ? 

I  have  two  quarts  of  chestnuts.  How  many  pint 
bowls  will  it  take  to  hold  them  ? 

Six  pints  of  oil  will  fill  how  many  quart  cans  ? 

What  part  of  a  quart  is  a  pint  ? 

If  a  quart  of  milk  costs  six  cents,  what  will  a  pint 
cost? 

If  a  pint  of  milk  ^osts  two  cents,  what  will  three 
pints  cost? 

Charlie  was  paid  a  cent  for  every  pint  of  cherries  he 
picked.  How  much  did  he  get  for  picking  three  quarts 
of  cherries  ? 


SEVEN  — 7. 


21 


How  many  barrels  are  there 
in  the  wagon  ? 

How  many  are  there  in  the  wagon 
and  on  the  ground  ? 

Six  barrels  and.  one  barrel  are  how  many  barrels  ? 

Name  the  days  of  the  week.     How  many  are  there? 

How  many  would  there  be  if  there  were  no  Satur- 
day? 

Seven  days  less  one  day  are  how  many  days  ? 

How  many  men  are  busy  with  the  apples  ? 

How  many  children  are  trying  to  helj)  ? 

Five  men  and  two  children  are  how  many  people  ? 

If  two  of  the  men  should  go  home  in  the  wagon,  how 
many  people  would  be  left? 

If  the  two  children  should  then  run  off  to  play,  how 
many  would  be  left  to  work  ? 

How  many  twos  are  there  in  three  ? 

How  many  twos  are  there  in  seven  ? 

There  were  seven  leaves  on  a  twig,  and  five  of  them 
were  blown  away.     How  many  were  left  ? 


22  ELEMENTS   OF   ARITHMETIC. 

Draw  a  triangle  A-     Draw  a  square  Q. 

How  many  sides  has  the  triangle  ? 

How  many  sides  has  the  square  ? 

How  many  sides  have  both  figures  ? 

How  many  more  sides  has  the  square  than  the  triangle  ? 

A  farmer  had  seven  tons  of  hay.  If  he  sold  three 
tons,  how  many  had  he  left? 

If  a  pair  of  boots  costs  three  dollars,  how  many  pairs 
can  you  buy  with  seven  dollars  ?  How  much  will  you 
have  left? 

A  boy  bought  seven  pencils.  He  fell  and  broke  four 
of  them.     How  many  whole  ones  had  he? 

Show  with  blocks  how  many  fours  there  are  in  seven. 

How  many  quart  measures  can  you  fill  with  seven 
pints  ?     What  part  of  a  quart  have  you  left  ? 

What  will  seven  peaches  cost,  if  each  one  costs  a  cent? 

This  is  the  figure  that  stands  for  seven  —  /. 

If  I  give  two  oranges  to  Lizzie  and  two  to  Annie, 
how  many  do  I  give  away  ? 

How  many  times  do  I  give  two  oranges  away? 

Two  times  two  oranges  are  how  many  oranges? 

Two  times  two  is  written  thus  :  2  X  2  =  4. 

What  does  the  sign  X  mean? 

Read  and  complete : 
3x2=?  1x5=?  4x1=?  3x1=? 

7x1=?  2x3=?  1x7=?  2x2=? 


SLATE  WORK. 

6  +  1=? 

7_2=?              4+?  =  7 

3x2=? 

1  +  6=^^ 

7_5=?              7-?  =  3 

2x2=? 

7-1=? 

5  +  2=?              3+?  =  7 

2x3=? 

7-6=?  2  +  5=?  7-?=4  7x1=? 


EIGHT  — 8. 


23 


10.  This  girl  has  poured  seven  pints  of  milk  into  the 
dish.  If  she  pours  in  one  pint  more,  how  many  pints 
will  then  be  in  the  dish  ? 

If  she  empties  the  dish  by  pouring  the  milk  into 
quart  cans,  how  many  cans  will  it  fill  ? 

After  she  has  filled  one  can,  how  much  is  left  in  the 
dish? 

How  many  arms  have  four  boys  ? 

How  many  horns  have  four  cows  ? 

How  many  chairs  must  I  put  with  six  chairs  to  make 
eight  chairs  ? 

Fanny  had  three  roses,  three  lilies,  and  two  pansieSo 
How  many  flowers  had  she  ? 

How  many  threes  are  found  in  eight?  How  much  is 
left? 

Edith  wrote  eight  Avords  on  her  slate,  and  rubbed 
out  three  of  them.     How  many  were  left? 

Five  birds  and  three  birds  are  how  many  birds  ? 

Three  geese  and  five  geese  are  how  many  geese  ? 

How  many  horses  can  be  shod  with  eight  horseshoes  ? 


•  •  •  • 


24  ELEMENTS   OF  ARITHMETIC. 

How  many  two-cent  stamps  can  be  bought  with  eight 
cents  ? 

What  two  equal  numbers  make  eight? 

What  four  equal  numbers  make  eight  ?     • 

A  gallon  contains  four  quarts.  How  many  gallons 
are  there  in  eight  quarts  ? 

This  figure  stands  for  eight  —  ^. 

Place  eight  blocks  in  two  equal  groups.  How  many 
are  there  in  each  group  ? 

What  is  half  of  eight  blocks  ? 

Eight  sheep  were  in  a  field,  and  half  of  them  got  out. 
How  many  were  left  ? 

Break  a  splint  into  halves.  Break  each  half  splint 
into  two  equal  parts.  Into  how  many  pieces  have  you 
broken  the  splint  ? 

Each  piece  is  called  a  fourth^  or  a  quarter. 

How  many  fourths  are  there  in  a  pie  ? 

Draw  a  square  on  your  slate.  Divide  it  into  four 
equal  parts.     Do  it  in  more  than  one  way. 

Divide  four  pears  into  four  equal  groups.  What  is  a 
fourth  of  four  pears  ? 

Place  eight  marbles  in  four  equal  piles.  What  part 
of  the  marbles  is  in  each  pile  ? 

Take  a  fourth  of  them  in  your  hand. 

A  fourth  of  eight  marbles  is  how  many  marbles  ? 

A  boy  who  had  eight  cents  spent  half  of  what  he  had  for 
paper.   How  much  did  he  spend  ?   How  much  had  he  left  ? 

A  boy  who  had  eight  miles  to  go  traveled  one  half  the 
distance  by  bicycle.     How  far  did  he  travel  by  bicycle  ? 

A  girl  who  earned  eight  dollars  per  month  paid  one 
half  of  it  for  her  necessary  expenses.  How  much  did 
she  save  ? 


EIGHT  — 8. 


25 


11.  Find  a  pint  measure  and  a  gill  measure. 
Which  is  the  smaller? 

Fill  the  gill  measure  with  water.     Pour  it  into  the 
pint  measure.     Do  this  until  you  fill  the  pint  measure. 
How  many  gills  of  water  fill  the  pint  measure  ? 
How  many  gills  are  there  in  two  pints  of  vinegar? 
How  many  pints  are  there  in  two  quarts  of  water  ? 
How  many  pints  are  there  in  three  quarts  of  water  ? 
In  eight  pints  of  water,  how  many  quarts  are  there  ? 


SLATE  WORK. 

2+6=? 

8-T=? 

8-5=? 

8-?  =4 

?  x2  =  8 

8x?  =  8 

2x4=? 

?-l  =  7 

?+5  =  8 

^  ?+3  =  8 

8-?=2 

4  +  4=? 

Add: 

2          5 

1 

4 

3 

4         6 

6          3 

7 

4 

2 

3          2 

How  much  is : 

4of4 

iof6 

iof4 

^of  2 

iof3 

iof8 

iof6 

iof  8 

J  of  2  X  2 

26 


ELEMENTS  OF   ARITHMETIC. 


ll^t^^^f'^ 


-i'^ftt 


12.   David  and  Fred  went  fishing. 

How  many  fishes  has  David,  the  boy  by  the  fence,  on 
his  string  ?     How  many  has  Fred  ? 

How  many  fishes  did  both  catch  ? 

How  many  more  did  David  catch  than  Fred  ? 

How  many  postal  cards  can  I  buy  with  nine  cents,  if 
they  cost  me  a  cent  apiece  ? 

Emma  had  a  bnncli  of  nine  roses,  but  two  of  them 
were  lost.     How  many  were  left? 

If  you  had  a  two-cent  piece,  how  much  more  would 
you  need  to  make  nine  cents  ? 

Carl  paid  three  cents  for  a  top,  three  cents  for  a 
pencil,  and  three  cents  for  a  marble.  How  much  did 
they  all  cost  him  ? 

Three  times  three  cents  are  how  many  cents  ? 

Divide  nine  plums  equally  among  three  girls. 

What  part  of  the  plums  will  each  receive  ? 

What  is  a  third  of  nine  pounds  of  sugar? 

There  are  six  boys  and  three  girls  in  a  class, 
many  children  are  there  in  the  class  ? 


How 


NINE  — 9.  27 

How  many  more  boys  are  there  than  girls  ? 

If  there  were  nine  eggs  in  a  nest  and  three  were 
taken  out,  how  many  were  there  left? 

Ethel  had  a  half-dime  and  two  two-cent  pieces.  How 
much  money  had  she  ? 

If  she  bought  oranges  at  four  cents  apiece  with  her 
money,  how  many  did  she  buy  ? 

How  much  shall  I  have  left  out  of  nine  dollars,  after 
paying  four  dollars  for  a  pair  of  boots  ? 

How  many  pints  are  there  in  nine  gills  ? 

How  many  quarts  are  there  in  nine  pints  ? 

Frank  wrote  his  name  nine  times  on  his  slate.  He 
wrote  it  well  only  seven  times.  How  many  times  did 
he  write  it  carelessly  ? 

How  many  cents  are  three  cents  and  six  cents  more  ? 

If  milk  costs  three  cents  a  pint,  how  many  pints  can 
I  buy  with  nine  cents  ? 

^ENi)  iwn)  (ct>ir) 


How  many  threes  are  there  in  nine  ? 
This  figure  stands  for  nine  —  ^. 

Here  are  some  shorter  ways  of  writing  the  number  of 
threes  in  nine : 

3'sin9  =  3;  or  3)  9  (3  ;  or  9-v-3  =  3. 

Read  and  complete : 

2'sin8=?  3'sin6=?  I'sin7=? 

4)8(?      6)6(?      1)5(?     9--3=?      8--8=  ?      9-i-l=? 


28  ELEMENTS  OF  ARITPIMETIC. 


13.   Count  the  balls  on  this  wire. 

What  two  equal  numbers  make  ten. 

Two  fives  are  how  many?     What  is  half  of  ten  balls? 

Ten  cents  make  a  dime.  How  many  cents  are  there 
in  half  a  dime  ? 

Jane  found  a  dime.  She  spent  half  a  dime  for  a 
doll.     How  much  had  she  left  ? 

How  many  dolls,  at  that  price,  could  she  have  bought 
with  the  dime  ? 

How  many  fingers  have  you? 

If  you  do  not  count  your  thumbs,  how  many  have  you  ? 

Ten  fingers  less  two  fingers  are  how  many  fingers  ? 

Place  ten  blocks  in  twos.  How  many  twos  do  you 
find  in  ten  blocks? 

At  two  cents  apiece,  what  will  five  bananas  cost  ? 

A  boy  who  had  ten  marbles  lost  eight  of  them.  How 
many  had  he  left  ? 

When  Carl  has  read  seven  pages,  how  many  pages 
more  must  he  read  to  make  ten  pages  ? 

Take  four  buttons  away  from  ten  buttons,  and  see 
how  many  are  left  ? 

Three  threes  and  how  many  make  ten? 

How  many  squares  can  you  make  with  ten  sticks  ? 

These  figures  stand  for  ten  —  fo. 

The  figure  on  the  right  is  called  zero.  What  is  the 
left-hand  figure  called  ? 

Write  the  figures,  /,  2,  d,  f ,  6,  6,  7,  8,  ^,  0, 


ONE  HALF  — ONE  THIRD  ^  ONE  FOURTH 


14.   What  do  you  call  each  part  of  this  circle  ? 

One  half  is  written  in  this  wa}^ :  ^. 

What  do  you  call  each  part  of  this  square  ? 

One  third  is  written  in  this  way :  ^. 

What  do  you  call  each  part  of  this  line  ? 


J.            X            ± 

1      "^       I       *       !      ^       I 

J. 
^    ,...1 

One  fourth  is  written  in  this  way : 

h 

Read  and  complete : 

iot2=?           loflO=? 

Jof6=? 

lof9=?            lof    4=? 

^of  8=? 

3  is  J  of?            2  is    lof? 

2  is  1  of? 

If  three  bananas  cost  nine  cents,  what  will  each  cost? 

/one\  /oneN  /oneA  /one\ /one\  /oneN  /one\/one\  /one\ 

\CENT/ IcentJ  Uemt/  [ciuy  \cimj  [cuirj  IcemtJ  I  cent/  I  cent  i 


What  part  of  nine  cents  are  three  cents  ? 

What  part  of  nine  cents  will  each  banana  cost  ? 

If  four  caps  cost  eight  dollars,  how  much  will  each  cost  ? 


What  part  of  eight  dollars  will  each  cap  cost? 


30 


ELEMENTS  OF   ARITHMETIC. 


15. 


REVIEW   EXERCISES. 


3  +  4=? 

6-4=? 

2x5=? 

8-4  = 

? 

6  +  2=? 

8-2=? 

3x3=? 

9-3  = 

? 

5  +  4=? 

10-5=? 

4x2=? 

6-2  = 

? 

2  +  7=? 

5-3=? 

1x7=? 

10-5  = 

? 

4  +  6=? 

9-6=? 

5x1=? 

7-^l  = 

? 

1  +  5=? 

7-1=? 

3x2=? 

10-2  = 

? 

lof  10=? 

7-5=    ? 

3+?=  5 

5x?  = 

10 

lof    6=? 

10-7=    ? 

4+?=   9 

?  x4  = 

8 

lof    9=? 

9-?=  5 

6-3=    ? 

9x1  = 

? 

lof    6=? 

?-3=  5 

10-?  =  10 

1  of  8  = 

? 

iof    4=? 

7+?=10 

8-?=  4 

iof  ?  = 

2 

iof    8=? 

?+3=  9 

9-9=    ? 

iof  ?= 

1 

2x   3=? 

4-4=? 

7-  4=? 

3+?  = 

=  9 

8+  2=? 

lof  2=? 

10-10=? 

8+?  = 

=  9 

lof  4=? 

8-4=? 

2x    2=? 

5+?  = 

=  9 

4-f-   2=? 

7x1=? 

6x   1=? 

10-?  = 

--2 

9-  5=? 

3  +  7=? 

6+  4=? 

10-?  = 

=  6 

10-  6=? 

7  +  2=? 

iof9=? 

10-?  = 

=  1 

Add:    2 

4            1 

3 

2 

4 

3 

1            2 

3 

3 

1 

1 

0            3 

1 

1 

2 

2 

3            4 

2 

4 

1 

From   9 

6            8 

7 

8 

5 

Take    3 

2            5 

3 

4 

3 

ELEVEN  TO  NINETEEN.  31 

16. 

One  ten. 


••••••••  • 

One  ten  and  one       are  eleven ;         written,  //. 


••••••••  •• 

One  ten  and  two       are  twelve  ;        written,  12. 

•  •••••  9^  •  •  • 

One  ten  and  three    are  thirteen ;      written,  /3. 

•••o««««  •••• 

One  ten  and  four      are  fourteen ;     written,  /^. 


One  ten  and  five       are  fifteen ;         written,  16. 

One  ten  and  six        are  sixteen ;        written,  /6. 

One  ten  and  seven   are  seventeen ;  written,  fj. 

One  ten  and  eight    are  eighteen ;     written,  /8. 

One  ten  and  nine      are  nineteen ;     written,  /^. 


32 


ELEMENTS  OF   ARITHMETIC. 


iPi|#ii/ ,  ,  ,„ 


.,;,/];, 


—  ■    -^-^^S^'      ^"^ 


17.   How  many  frogs  are  there 
in  the  pond? 
If  the  one  on  the  edge  should  jump  in,  how 
many  would  there  be  in  the  water  ? 

How  many  ones  are  there  in  eleven  frogs  ? 

How  many  tens  are  there  in  eleven  frogs  ? 

There  were  eleven  trees  in  a  garden.  Two  of  them 
were  peach  trees  and  the  rest  were  pear  trees.  How 
many  pear  trees  were  there  ? 

What  is  the  greatest  number  of  two-cent  pieces  you 
can  have  in  eleven  cents  ? 

A  week  and  four  days  are  how  many  days  ? 

How  many  horses  can  be  shod  all  around  with  eleven 
shoes  ?     How  many  shoes  are  left  ? 

Harry  had  three  pieces  of  money  which  made  eleven 
cents.     What  were  they? 

If  he  should  change  them  for  two  pieces  of  money, 
what  coins  would  they  be  ? 

He  took  a  ride  upon  a  street  car  that  cost  him  five 
cents.     How  much  had  he  left  ? 

There  are  six  boys  and  five  girls  in  the  first  clasSc 
How  many  pupils  are  there  in  the  class  ? 


ELEVEN— 11.  33 

Mr.  Snow  had  eleven  cows.  Seven  were  in  the 
pasture  and  the  others  were  in  the  barn.  How  many 
were  in  the  barn? 

This   figure   is   a  pentagon.     How  many 
sides  has  it  ? 

How  many  pentagons  can  you  make  with 
eleven  sticks  ? 

Make  a  story  about  nine  nuts  and  two  nuts. 

Make  a  story  about  eleven  dogs  less  ten  dogs. 

Two  squares  and  a  triangle  have  how  many  sides  ? 

Two  triangles  and  a  square  have  how  many  sides  ? 

There  are  six  boys  in  my  class  to-day.  When  they 
are  all  present,  there  are  eleven.    How  many  are  absent? 

David's  father  has  eleven  horses  and  Henry's  has  five 
less.     How  many  horses  has  Henry's  father  ? 

A  writing  book  and  a  pencil  cost  together  eleven 
cents.  The  pencil  cost  two  cents.  How  much  did  the 
writing  book  cost  ? 

SLATE  WORK. 

10+   1=?  11-3=?  6+  5=?  3+?  =  ll 

11-   1-?  8  +  3=?  5+   G=?  11-?=   7 

11-10=?  11-8=?  11-6=?  ?+9  =  ll 

11^  1_?  3  +  8=?  11-   5=?  5     6     11 

11^10=?  11-3=?  11-f-  5=?  11     2       9 

9+   2=?  11-8=?  11-   6=?  11-?=  5 

2+9=?  T  +  4=?  11-11=?  ?-8=3 

11-   2=?  11-4=?  11-   0=?  ?+T  =  ll 

11-  9=?  11-7=?  llx   1=?  G+?  =  ll 

11h-   2=?  4  +  7=?  1x11=?  3  +  7=    ? 

11-  9=?  4)11(?  11-11=?  10-?=  6 

EL.    OF    AR.  3 


34 


ELEMENTS    OF    ARITHMETIC. 


^^\Vf   />V)^]T        18.   How  many  eggs  are  there  in 
/l  '      ^-  'this  nest? 

AVhat  other  name  is  given  to  twelve  eggs? 
If  you  should  buy  a  dozen  oranges,  how  many 
would  you  get  ? 

If  they  should  cost  you  a  cent  apiece,  how  much  would 
you  pay  for  them  ? 

If  you  should  put  half  of  them  in  a  dish,  how  many 
would  there  be  in  the  dish?  How  many  others  would 
there  be  ? 

What  is  another  name  for  six  oranges,  or  six  of 
anything  ? 

There  are  six  working  days  in  a  week.  How  many 
working  days  are  there  in  two  Aveeks  ? 

A  baby  had  twelve  spools  to  ]Dlay  with,  and  one  rolled 
away.     How  many  were  left  ? 

If  it  takes  ten  yards  of  lace  to  trim  a  dress  and  two 
yards  to  trim  an  apron,  how  much  will  be  needed  to 
trim  both  ? 

I  have  six  two-cent  pieces.     How  much  money  have  I? 

How  many  two-cent  stamps  can  I  buy  with  a  dime 
and  two  cents  ? 


TWELVE  — 12.  35 

James  bought  a  dozen  bananas.  He  ate  three,  gave 
three  to  his  brother,  and  three  to  his  sistero  How  many- 
had  he  left? 

How  many  threes  are  there  in  twelve  ? 

Place  twelve  beans  in  groups  of  three  beans  each. 

Into  how  many  groups  have  you  separated  the  twelve 
beans  ?     What  part  of  the  whole  is  each  group  ? 

What  is  a  fourth  of  twelve  miles  ? 

How  many  months  are  there  in  a  year  ?     Name  them. 

If  Willie  is  out  of  school  during  June,  July,  August, 
and  September,  how  many  months  in  the  year  does  he 
attend  school  ? 

Three  horses  need  how  many  shoes? 

What  part  of  twelve  shoes  does  each  horse  need  ? 

If  the  ground  is  covered  with  snow  during  a  third  of 
the  year,  how. many  months  is  it  covered?  How  many 
months  in  the  year  is  the  ground  bare  ? 

Five  plants  in  the  room  are  in  blossom,  and  seven  are 
not.     How  many  plants  are  there  in  the  room  ? 

Take  a  foot  rule.  Measure  a  foot  on  your  desk. 
Measure  the  width  of  a  table.  See  hoAv  your  rule  is 
divided.  What  is  each  of  the  larger  parts  of  a  foot 
called?     How  many  inches  are  there  in  a  foot? 

If  I  had  a  piece  of  string  a  foot  long  and  cut  off  four 
inches,  how  many  inches  would  be  left  ? 

What  number  of  inches  must  I  add  to  seven  inches 
to  make  a  foot  ? 

How  many  inches  long  is  a  strap  that  is  just  one  half 
of  a  foot  in  length  ? 

Six  cows  have  how  man}^  horns? 

Each  of  three  windows  has  four  panes  of  glass.  How 
many  panes  of  glass  are  there  in  the  windows  ? 


36  ELEMENTS  OF   ARITHMETIC, 


SLATE  WORK. 


11+  1=? 

5  +  3  = 

=,  •? 

12-4=? 

12-6=? 

12-    ?  =  11 

9+?  = 

=  12 

3x4=? 

?-7  =  5 

? +11  =  12 

3  +  9  = 

=  2 

~i 

Lof  12=? 

ll-?  =  8 

12-11=? 

?  -9  = 

=  3 

7+?=12 

?-6  =  Q 

?+   2  =  12 

4x3  = 

=  ? 

12-7=? 

4  +  6=? 

2  +  10=? 

12-3  = 

=  ? 

?-5  =  7 

11-7=? 

12-    ?  =  10 

iof  12  = 

=  ? 

5  +  7=? 

9-6=? 

?-10  =  2 

8+?  = 

=  12 

6-\-6=? 

3x3=? 

6x    2=? 

12-8  = 

~  ? 

12-?  =  6 

?+4  =  12 

12-  2=? 

4  +  8  = 

=  ? 

Lof  12=? 

8  +  3=? 

12-  5=? 

12-?  = 

=  8 

^  2x?  =  12 

10-7=? 

8+  4=? 

?x3= 

=  12 

12-7=? 

?  +9  =  12 

12-    ?=7 

5+?  = 

=  12 

12-8=? 

?  -8  =  4 

8+    ?  =  12 

12-5  = 

=  ? 

12-4=? 

?  x6  =  12 

12-  3=? 

4x?= 

=  12 

?+5  =  12 

3+?  =  12 

12-    ?  =  2 

7  +  5  = 

-  9 

?+6  =  12 

?-7  =  5 

Add: 

4              1 
3             5 

2             4 

3 
4 
4 

1 

2 
3 

2 
3 
4 

3  6 

4  1 
3             4 

From         11 

9 

10 

11 

10           11 

Take           8 

5 

7 

5 

2            7 

Six       2's  =  ? 

Two 

6's  = 

:? 

Two    3's=: 

=  ?       2)6(? 

Three  4's  =  ? 

Four 

3's  = 

:? 

Three  2's  = 

=?       4)12(? 

Five     2^s  =  ? 

Three  3's  = 

:? 

Four    2's=: 

=?        ?)12(2 

THIRTEEN  — 13. 


87 


19.   Count  the  number  of  stripes  in  the  flag. 

What  colors  are  the  stripes  ?  How  many  red  ones  are 
there  ?     How  many  Avhite  ones  ? 

If  one  of  the  red  ones  should  be  torn  off,  how  many 
stripes  would  be  left  ? 

Byron  placed  eleven  blocks  in  one  group  and  two  in 
another.     How  many  blocks  were  there  in  both  groups? 

If  he  should  put  one  of  the  eleven  blocks  with  the 
other  group,  how  many  would  there  be  in  both  groups? 
How  many  in  each  group  ? 

What  is  the  largest  piece  of  money  that  can  be  used 
in  paying  thirteen  cents  ? 

How  much  more  is  needed  to  make  thirteen  cents  ? 

I  had  thirteen  cents,  and  I  spent  four  cents  for  stampSc 
How  many  cents  had  I  left  ? 

Four  two-cent  stamps  and  a  five-cent  stamp  cost  how 
much  ? 

I  walked  six  miles  yesterday  and  seven  miles  to-day. 
How  many  miles  did  I  walk  in  all  ? 


38 


ELEMENTS  OF  ARITHMETICo 


SLATE  WORK. 

Supply  the  Signs o 

7  +  6  = 

? 

10  + 

3=? 

13-?=  6 

3     4     12 

13-6  = 

'? 

13- 

3=? 

13-?=   2 

7     2       9 

13-7  = 

? 

13- 

10=? 

8+?  =  13 

11     5       6 

6  +  7  = 

? 

3  +  10=? 

3+?  =  13 

8     2       4 

8  +  5  = 

? 

11  + 

2=  ? 

13-?=   4 

4     2       8 

13-8  = 

? 

13- 

2=  ? 

13-?  =  10 

4     2       6 

13-5  = 

? 

13- 

11=? 

13-?=   8 

?  +  ?  =  13 

5  +  8  = 

? 

2  +  11=? 

9+?  =  13 

?-5=  5 

9  +  4  = 

? 

13x 

1=? 

2+?  =  13 

?-?=   9 

13-4  = 

? 

Ix 

13=? 

5+?  =  13 

?  X  ?=  9 

13-9  = 

? 

13- 

1=? 

13-?  =  12 

?  —  ?=   2 

4  +  9  = 

? 

13- 

13=? 

13-?=   7 

?  +  ?  =  12 

Acid: 

2 

8 

8 

5             4 

6       •     2 

5 

4 

2 

6              2 

4            5 

6 

5 

3 

1              6 

3            6 

5 

6 

7 

5              6 

5            4 

5 

3 

4 

5             5 

8            4 

2 

2 

2 

1             2 

0            4 

From 

13 

13 

13            12 

13          12 

Take 

8 

2 

6            10 

9            7 

From 

12 

13 

13            13 

12          13 

Take 

5 

7 

4             5 

9            3 

FOURTEEN  — 14. 
2  3 


39 


•  •  ••  ••••  •• 


20.  How  many  dots  are  there  in  each  of  these  figures? 

What  two  numbers  malve  fourteen  in  the  first  figure  ? 
In  the  second  ?     In  the  tliird  ?     In  the  f ourtli  ? 

How  many  days  are  there  in  two  weeks  ? 

How  many  more  days  are  there  in  two  wrecks  than  in 
one  week? 

At  seven  cents  apiece,  how  many  tops  can  I  buy  with 
fourteen  cents  ? 

Fourteen  geese  went  down  to  the  pond.  Seven  of 
them  soon  came  back.    How  many  remained  at  the  pond  ? 

With  your  blocks,  count  by  twos  to  fourteen. 

Begin  with  one  block,  and  count  by  twos  to  thirteen. 

I  have  fourteen  cents  in  two-cent  pieces.  Hoav  many 
tv.^o-cent  pieces  have  I  ? 

If  you  had  a  dime,  how  many  more  cents  would  it 
take  to  make  fourteen  cents  ? 

How  many  cents  do  you  need  to  put  with  a  half-dime 
to  make  fourteen  cents  ? 

How  many  pints  are  there  in  seven  quarts  ? 

How  many  quarts  are  there  in  fourteen  pints  ? 

A  foot  and  two  inches  are  how  many  inches  ? 

Kobert  sharpened  fourteen  j^ei^cils  for  me.  The 
points  of  two  of  them  w^ere  broken.  How  many  Avere 
left  with  good  points  ? 

I  have  two  three-cent  pieces.  How  many  more  cents 
will  it  take  to  make  fourteen  cents  ? 

Marie  is  8  years  old  to-day.  In  how  many  years  will 
she  be  14  ? 


40  ELEMENTS  OF  ARITHMETIC. 


SLATE 

WORK. 

2x7=? 

7  + 

?  =  14 

Supply  the  Signs. 
5     9     14 

?  +  ?=14 

6  +  8=? 

?  _ 

10=  4 

14     8       6 

?-?  =  10 

14-9=? 

?  + 

5  =  14 

12     2     14 

?  X  ?=14 

iofl4  = 

?     14- 

?  =  11 

14     4     10 

?-?=  7 

7)14(? 

14- 

?=   2 

14     7       7 

?  +  ?  =  12 

7x.2=? 

5x 

2=    ? 

12-6=    ? 

11-?=    ? 

2)14(? 

1)14(? 
14)14(? 
14x1=? 

?  X 

4x 

'3x 
3x 

2  =  12 

2=    ? 
?=   6 
3=    ? 

12-4=    ? 
12-3=    ? 
12  —  2=    ? 
12-1=    ? 

12—?=    ? 
13-?=    ? 
14-?=    ? 
10-?=    ? 

Add: 

2  5 

3  4 

7            5 

3 

7 
4 

4 
3 

7 

1            6 

3            2 
8  •          6 

2            3 
7            4 
5            7 

From 
Take 

14 

8 

13 
5 

14           12 

10             7 

14           13 

9             8 

21.  Take  your  foot  rule  and  draw  a  line  on  the  board 
just  as  long  as  the  rule.     How  long  is  the  line  ? 

Add  to  this  line  another  of  the  same  length,  and 
then  another.     How  long  is  the  whole  line  now  ? 

Take  the  yard-stick  and  measure  the  line  with  it. 
How  does  it  compare  in  length  with  a  yard  ? 

How  many  feet  are  there  in  a  yard  of  anything  ? 

Measure  off  a  yard  on  the  edge  of  the  table. 

What  is  always  measured  by  the  yard  ? 

How  many  feet  are  there  in  two  yards  of  string  ? 

A  room  twelve  feet  long  will  need  how  many  yards 
of  carpet  to  extend  the  length  of  the  room  ? 


FIFTEEN  — 15.  41 


22.  How  many  rows  of  squares 
are  there  in  this  figure  ? 

How  many  squares  are  there  in 
each  row? 

How  many  squares  are  there  in 
the  three  rows  ? 

Tell  from  the  picture  how  many  threes  make  fifteen  ? 
How  many  fives  ? 

I  have  a  dime  and  a  half-dime.     For  how  many  cents 
can  I  exchange  them  ? 

Look  at  the  figure  given  above  and  tell  me  what  a 
third  of  fifteen  squares  is. 

In  fifteen  feet,  how  many  yards  are  there  ? 

There  were  fifteen  bananas  in  a  bunch.     After  ten 
were  sold  how  many  were  left  ? 

What  three  pieces  of  money  make  fifteen  cents  ? 

If  a  pound  of  beef  costs  ten  cents,  what  will  a  pound 
and  a  half  cost  ? 

Count  by  threes  to  fifteen. 

Write  on  your  slate  in  a  vertical  line  enough  threes 
to  make  fifteen.     How  many  did  you  write  ? 

Five  triangles  have  how  many  sides  ? 

A  dozen  and  three  oranges  are  how  many  oranges? 

Johnny  is   fifteen  months   old.     Hoav  many  months 
more  than  a  year  is  that  ? 

I  bought  three  oranges  for  fifteen  cents.     How  much 
did  I  pay  for  each  orange  ? 

Three  times  what  number  equals  fifteen  ? 

Five  times  what  number  equals  fifteen  ? 

A  boy  ran  on  three  errands,  receiving  five  cents  for 
each.     How  much  did  he  earn  ? 

How  many  eggs  are  a  dozen  and  a  quarter  eggs? 


42 


ELEMENTS  OF   ARITHMETIC. 


23.  Make  a  figure  like  the  one  on  the  previous  page 
without  the  two  inside  horizontal  lines. 

Into  how  many  equal  parts  is 
this  figure  divided? 

Each  of  these  parts  is  called  a 

fifth. 

How  many  fifths  are  there  in 
the  figure  which  you  drew  ? 

How  many  fifths  are  there  in  any  figure  ?     Li  any  line  ? 

If  a  pie  were  divided  into  five  equal  parts  and  you 
were  given  one  of  the  parts,  what  part  of  the  pie  would 
you  receive  ? 

Now  you  may  draw  the  two  inside  horizontal  lines 
as  shown  in  the  previous  figure.  How  many  squares 
have  you  now  made  ? 

How  many  squares  are  there  in  a  fifth  of  the  figure  ? 

What  is  a  fifth  of  fifteen  sheep  ? 

Separate  ten  blocks  into  five  equal  parts.  What  is  a 
fifth  of  ten  blocks  ? 

This  is  the  way  we  write  a  fifth :  \. 


SLATE  WORK. 

10+  5=? 

15-  3=? 

T+?  =  15 

15-?=  5 

15-   5=? 

5x    3=? 

9+?  =  15 

15-?=  3 

15-10=? 

iofl5=? 

15-?  =  11 

15-?=  1 

5  +  10=? 

15-  3=? 

15-?=  5 

15-?  =  15 

3x   5=? 

15-12=? 

13+?  =  15 

3x  ?  =  15 

iofl5=? 

12+   3=? 

15-?=   8 

5x  '.^  =  15 

15-f    5=? 

3  +  12=? 

8+?  =  15 

Ix  ?  =  15 

SIXTEEN  — 16. 


43 


24.   Count  the  plates  in  these  two  piles. 

How  many  are  there  in  each  pile  ? 

How  many  are  there  in  both  piles  ? 

If  these  plates  were  in  two  equal  piles,  how  many 
would  there  be  in  each  ? 

Take  sixteen  pieces  of  paper  and  imagine  that  they 
are  plates.  Place  them  in  two  equal  piles.  What  is 
one  half  of  sixteen  plates  ? 

There  were  sixteen  children  in  a  class.  One  stormy 
day  one  half  of  them  were  absent.  How  many  were 
present  ? 

How  many  shoes  do  four  horses  need  ? 

How  many  oxen  can  be  shod  with  sixteen  shoes,  if 
each  ox  requires  eight  shoes  ? 

With  your  blocks  count  by  twos  to  sixteen. 

Write  on  your  slates  in  a  vertical  line  a  sufficient 
number  of  twos  to  make  sixteen.  How  many  twos  are 
there  in  sixteen  ? 

Begin  with  sixteen  and  count  backward  by  twos. 

There  are  four  quarts  in  a  gallon.     Prove  it. 


44  ELEMENTS   OF   ARITHMETIC. 

How  many  quarts  are  there  in  two  gallons? 
How  many  quarts  are  there  in  three  gallons  ? 
In  sixteen  quarts,  how  many  gallons  are  there  ? 
What  part  of  a  gallon  is  a  quart  ? 
How  many  pints  are  there  in  a  gallon  ? 


SLATE 

WORK. 

4x4=    ? 

iofl6=    ? 

16-   9  =  ? 

3  +  ?  =  16 

8x2=    ? 

I  of  16=    ? 

16-   4  =  ? 

.  ?-8=   8 

2x?  =  lG 

8+?  =  16 

16-10  =  ? 

?  +  5  =  16 

16-^2=    ? 

16-7=    ? 

16-12  =  ? 

?-h9  =  16 

16-4=    ? 

?  +  4  =  16 

16-  5  =  ? 

16-?=  5 

16h-8=    ?  10  +  6=    ?         16-   6  =  ?        16-?=   3 

Add: 


3 

4 

8 

5 

6 

4 

4 

5 

2 

6 

3 

2 

2 

3 

2 

3 

2 

3 

7 

3 

4 

2 

4 

4 

Subtract : 

16 

16 

15 

15 

16 

16 

10 

15 

8 

4 

9 

5 

Answer  instantly : 

7+  8 

15- 

-5 

lof  16 

3x    3 

9+  4 

16- 

^1 

lof 

8 

4x   4 

8-  5 

16- 

-8 

lof 

4 

2x    2 

16-   9 

16- 

-4 

lof 

2 

3+   8 

14-11 

16- 

-2 

1  of  14 

8+   8 

12+  4 

14- 

^2 

lof 

6 

5+  8 

3x   4  10-2  lof  12  11-   7 

8x    2  12-2  16-12  13-  6 


ONE   EIGHTH.  45 


25.   Into  how  many  parts  is  this  line  divided?     How 
do  these  parts  compare  in  size  ? 

Each  of  these  equal  parts  is  called  an  eighth. 

How  many  eighths  are  there  in  this  square  ? 

Point  to  an  eighth  of  this  square. 

An  eighth  is  written  in  this  way  —  -J. 

Separate  eight  blocks  into  eight  equal  parts. 

What  is  an  eighth  of  eight  blocks  ? 

If  you  should  divide  sixteen  pears  equally 


among  eight  boys,  what  part  of  them  would  each  boy 
get? 

How  many  would  each  boy  get  ? 

What  is  an  eighth  of  sixteen  horses  ? 

A  farmer  had  sixteen  tons  of  hay.  If  he  should  sell 
an  eighth  of  it,  how  many  tons  would  he  have  left  ? 

Draw  a  square.  Divide  it  into  halves.  Now  divide 
your  square  into  four  equal  parts. 

What  is  each  part  called? 

How  many  fourths  are  there  in  the  square  ? 

How  many  fourths  are  there  in  half  of  the  square  ? 

What,  then,  is  another  name  for  half  of  the  square  ? 

What  other  name  might  we  give  to  two  fourths  of  a 
pie? 

Now  divide  this  square  into  eight  equal  parts.  What 
is  each  part  called  ? 

How  many  eighths  are  there  in  the  square  ? 

How  many  eighths  are  there  in  half  of  the  square  ? 

How  many  eighths  are  there  in  a  fourth  of  the  square  ? 

Tell  me  all  the  names  you  know  for  a  half  of  anything. 

Give  me  one  other  name  for  a  fourth  of  anything. 


46 


ELEMENTS  OF  ARITHMETIC. 


•••  ••  •••• 

••  ••  •••• 

•  •    •  •    e    • 

•  •   •  •    •    • 


26.    Copy  these  figures  on  your  slate. 

Fill  in  the  blank  half  of  each  with  enough  dots  to 
make  seventeen  in  each  one. 

Name  three  pieces  of  money  that  will  make  seventeen 
cents  o 

There  were  seventeen  pine  trees  in  a  field.  If  seven 
of  them  were  chopped  down,  how  many  were  left  stand- 
ing? 

In  front  of  my  house  there  are  nine  maple  trees  on 
one  side  of  the  street  and  eight  on  the  other.  How 
many  are  there  on  both  sides  ? 

It  is  seventeen  miles  from  one  station  to  the  next. 
When  the  train  has  gone  eleven  miles  from  one  station, 
how  many  miles  has  it  to  go  before  it  reaches  the  other  ? 

A  house  has  six  windows  in  one  side,  seven  in  another, 
and  two  in  each  end.  How  many  windows  are  there  in 
the  house? 

A  man  earned  seventeen  dollars  a  week  and  spent 
twelve  dollars  of  it.     How  much  had  he  left  ? 

Mr.  White  sold  his  turkeys  for  seventeen  dollars. 
How  many  sheep  at  five  dollars  apiece  could  he  buy 
with  the  money  ?     How  much  would  he  have  left  ? 

A  newsboy  who  bought  seventeen  papers  sold  all  but 
five  of  them.     How  many  did  he  sell  ? 

Two  boys  start  from  the  same  place.  One  walks  eight 
miles  in  one  direction,  the  other  walks  nine  miles  in  the 
other  direction.     How  far  are  they  apart  then  ? 


SEVENTEEN  — 17. 


47 


SLATE  WORK. 

17-   2=? 

12-f-?-17 

17^  5=? 

15  =  17-? 

IT-   8  =  ? 

17-?=  7 

17- 

-10  =  ? 

17  =  10-f? 

17-f-  2=? 

17-?  =  13 

17- 

-   8  =  ? 

17  =  2x8-h? 

17-^  3  =  ? 

?  +  8  =  17 

17- 

-  6  =  ? 

17  =  3x5-f? 

17-f-  4  =  ? 

6-f-?  =  17 

17- 

-   9  =  ? 

17  =  7x2-f? 

7-^10  =  ? 

17-?  =  10 

17- 

-  7=? 

17-4x3-f? 

Add: 

2 

5          4 

3           6 

9 

4 

3 

4          8 

7 

2 

1 

3 

Add: 

12 

5 


15 

2 


7 
10 


6 
11 


Answer  instantly  : 


9-1-  4 

7-f  8 
6-fll 

4-f-  8 

5-f  9 

9h-  8 

6+  4 


12-  5 
15-  8 
17-  9 
16-12 
11-  5 

13-  8 

14-  7 


7x2 
4x3 
3x3 
6x2 
4x4 
2x5 


5 
12 


4 
13 


9 

8 

9 

6 

7 

5 

8 

6 

5 

4 

8 

6 

5 

8 

Oat  of 

17 

14 

15 

16 

17 

17 

Take 

8 

8 

10 

7 

16 

3 

1)16( 
2)16( 
4)16( 
8)16( 
16)16( 
5)15( 
7)14( 


48 


ELEMENTS   OF   ARITHMETIC. 


:^^^==-^ 

h-#2-          j»r^^:^                       — "      '^^^^mr^ 

°^======:^— — --— — ^^                                                                         1\ 

1 

27.   How  many  birds  are  resting  on  these  wires? 

How  many  are  there  in  each  group  ? 

If  one  from  the  larger  group  should  join  the  smaller 
group,  how  many  would  there  be  in  each  group  then  ? 

There  are  eighteen  windows  in  a  house.  If  ten  of 
Miem  are  in  the  sides  of  the  house,  how  many  are  there 
in  the  ends  ? 

Mary's  rose-bush  had  eighteen  rose-buds  on  it,  and 
she  picked  twelve  of  them.  How  many  rose-buds  were 
left  on  the  bush? 

Lizzie  made  three  button-holes  each  day  for  a  week, 
beginning  on  Monday.  How  many  had  she  made  by 
Saturday  night? 

At  three  cents  apiece,  how  many  peaches  can  I  buy 
with  eighteen  cents? 

Fred  learned  six  verses  of  a  poem  each  day  for  three 
days.     How  many  verses  did  he  learn  ? 

What  part  of  the  eighteen  verses  did  he  learn  each 
day? 

A  dozen  and  a  half  eggs  are  how  many  eggs  ? 

If  lace  is  worth  a  cent  an  inch,  what  will  eighteen 
inches  of  it  cost? 


ONE   SIXTH. 


49 


28.   Into  how  many  parts  is  this  figure  divided  ? 

Each  part  is  called  a  sixth. 

How  many  sixths  are  there  in  it? 

How  many  sixths  are  there  in  a  pie  ?     In  a  melon  *? 

A  sixth  is  written  in  this  way  —  -J . 

Make  a  picture  of  a  pie  and  divide  it  into  sixths.    How 
many  sixths  are  there  in  half  a  pie  ?     In  one  third  ? 

If  you  cut  a  card  that  is  six  inches  long  into  sixths, 
how  long  will  each  piece  be  ? 

What  is  a  sixth  of  twelve  words  ? 

John's  overcoat  cost  eighteen  dollars.     His  hat  cost 
one  sixth  as  much.     What  did  his  hat  cost  ? 

Draw  a  line.     Divide  it  into  halves,  and  then  into 
sixths. 

Draw  another  line  of  the  same  length.    Divide  it  into 
thirds,  and  then  into  sixths. 

How  many  sixths  are  there  in  half  of  the  line  ? 

How  many  sixths  are  there  in  a  third  of  the  line  ? 

One  half  is  how  many  fourths  ?     How  many  sixths  ? 
How  many  eighths  ? 

Express  with  figures  one  sixth,  two  sixths,  five  sixths, 
three  sixths. 

Write  all  the  expressions  that  you  know  which  mean 
the  same  as  one  half. 

Write  all  the  expressions  that  you  know  which  mean 
the  same  as  one  fourth. 

How  many  sixths  are  1  sixth  and  3  sixths  ? 
How  many  eighths  are  3  eighths  and  4  eighths  ? 

EL,    OF   AR.  — ^4 


50  ELEMENTS  OF   ARITHMETIC. 


SLAT! 

:   WORK. 

18-2  =  ? 

6X?: 

=  18 

18-   6. 

=  ? 

iof  16  =  ? 

18^3  =  ? 

3X?: 

=  18 

18-   9. 

=  ? 

lof  16  =  ? 

18-6  =  ? 

9X?: 

=  18 

18-12  = 

=  9 

iof  16  =  ? 

18-9  =  ? 

2X?: 

=  18 

18-15  = 

=  9 

iof  15  =  ? 

-J- of  18=? 

9+?: 

=  18 

18-  7  = 

_  9 

iof  12  =  ? 

iof  18  =  ? 

10+?: 

=  18 

18-   3  = 

=3     ? 

Iof    6  =  ? 

iof  18  =  ? 

8+?  = 

=  18 

18-   5  = 

33     ? 

Iof    9  =  ? 

12  +  ?  =  18 

9  +  8 

=  2 

iof?  = 

=  2 

11-7=? 

18-?=   2 

14-6 

=  9 

iof?  = 

=  6 

8  +  4  =  ? 

6  +  ?  =  18 

6  +  9: 

-7 

lof?  = 

=  4 

14-7  =  ? 

18-?  =  14 

8  +  5: 

=2 

lof?  = 

=  3 

6x3  =  ? 

18-?  =  10 

b  +  6. 

=  9 

lof?  = 

=  5 

4  +  9  =  ? 

7+?=18 

9  +  7: 

=  ? 

4of?= 

=  1 

1  of  15  =  ? 

Add: 

4            5 

2 

3 

6 

7 

6 

3            2 

9 

7 

4 

1 

3 

6            T 

2 

2 

5 

4 

4 

5           4 

4 

4 

3 

5 

4 

3            4 

5 

4 

5 

5 

6 

6            6 

8 

1 

1 

2 

2 

7            2 

2 

3 

8 

7 

5 

2            3 

1 

6 

4 

2 

1 

Subtract : 

18 

18 

18 

18 

18 

18 

17 

10 

14 

9 

7 

5 

ISmSTETEEN  — 19. 


51 


29.  How  many  doves  do 
you  see  in  this  picture  ? 

What  two  numbers  make 
nineteen  ? 

Take  nineteen  blocks.  I 
will  name  one  part  of  nine- 
teen, and  you  may  find  out 
the  other  part  with  your 
blocks:  12;  6;  8;  15;  5; 
16;  7;  9;  11;  4;  13;  10; 
14;  3;  17. 

Two  weeks  and  five  days 
are  how  many  days  ? 

Alice  is  nineteen  years  old. 
How  old  was  she  ten  years 
ago? 

There  were  nineteen  cars  at 
the  station.  After  an  engine 
had  drawn  away  twelve  of 
them,  how  many  were  left? 

Out  of  nineteen  pupils,  five 
were  absent.  How  many 
were  present? 

In  a  house  there  are  nine- 
teen rooms.  If  eight  are  up- 
stairs, how  many  are  down- 
stairs ? 

How  many  pairs  of  boots  at  two  dollars  a  pair  can  I 
buy  with  nineteen  dollars  ?    How  much  shall  I  have  left  ? 
How  many  triangles  can  you   make  with   nineteen 
sticks  ?     How  many  sticks  will  be  left  ? 
How  many  are  nineteen  less  ten? 


52  ELEMENTS   OF   ARITHMETIC 

How  many  gallon  cans  can  be  filled  with  nineteen 
quarts  of  milk  ?     How  many  quarts  will  be  left  ? 

How  many  five-cent  stamps  can  you  buy  with  nine- 
teen cents  ?     How  many  cents  will  be  left  ? 


SLATE 

WORK. 

10  +  9  =  ? 

19- 

-9  =  ? 

13  +  ?  =  19 

19  =  3x5+? 

12  +  7=? 

19- 

-4  =  ? 

19-?  =  10 

19  =  4x4  +  ? 

19-8  =  ? 

19- 

-7=? 

11  +  ?  =  19 

19  =  5x2+? 

19-6  =  ? 

19- 

-5  =  ? 

19-?  =  14 

19  =  3x6+? 

14  +  5  =  ? 

19- 

-8  =  ? 

8  +  ?  =  19 

19  =  7x2+? 

19-7  =  ? 

19- 

-6  =  ? 

19-?=   9 

19  =  2x9  +  ? 

Answer  ins 

jtantly : 

3x6 

8  +  9 

2x8 

lofl2 

15-^5 

5  +  9 

2x5 

lof    8 

13-6 

6  +  9 

2x3 

-1-  of  18 

lofl8 

4  +  9 

2x9 

i-ofl6 

7  +  8 

11  +  9 

2x7 

lof  10 

5x3 

15-9 

2x4 

iofl6 

14-7 

19-9 

2x6 

iof    8 

17-9 

14-9 

.3x5 

i  of  12 

iofl2 

18-9 

3x2 

4  of    8 

8  +  6 

12-9 

3x4 

iofl6 

Add: 

2 

6 

3 

4 

3 

5 

3 

4 

5 

4 

3 

T 

5 

2 

6 

4 

3 

3 

4 

3 

4 

4 

3 

2 

5 

4 

1 

3 

6 

2 

REVIEW   EXERCISES.  63 

30.  REVIEW   EXERCISES. 


1  +  5=? 

11  +  5=? 

9-4  =  ? 

19-4=? 

3  +  6  =  ? 

13  +  6  =  ? 

9-8  =  ? 

19-8=? 

5  +  4-? 

15  +  4  =  ? 

7-5=? 

17-5=? 

2  +  6  =  ? 

12  +  6  =  ? 

7-3=? 

17-3  =  ? 

4  +  4  =  ? 

14  +  4  =  ? 

5-4=? 

15-4=? 

6  +  2  =  ? 

16  +  2  =  ? 

5-2  =  ? 

15-2  =  ? 

3  +  4  =  *^ 

13  +  4  =  ? 

8-7  =  ? 

18-7  =  ? 

7  +  2  =  ? 

17  +  2  =  ? 

8-5  =  ? 

18-5  =  ? 

4  +  2  =  ? 

14  +  2=? 

6-4  =  ? 

16-4  =  ? 

5  +  3  =  ^ 

15  +  3  =  ? 

6-1  =  ? 

16-1=? 

2  +  7  =  ? 

12  +  7  =  ? 

4-3  =  ? 

14-3  =  ? 

3  +  5=? 

13  +  5  =  ? 

4-2  =  ? 

14-2  =  ? 

4  +  3  =  ? 

14  +  3  =  ? 

8-4  =  ? 

18-4  =  ? 

5  +  1  =  ? 

15  +  1  =  ? 

8-6  =  ? 

18-6  =  ? 

1  +  8  =  ? 

11  +  8  =  ? 

7-6  =  ? 

17-6  =  ? 

2  +  4  =  ? 

12  +  4  =  ? 

7-4  =  ? 

17-4=? 

3  +  2  =  ? 

13  +  2  =  ? 

7-7  =  ? 

17-7=? 

4  +  1  =  ? 

14  +  1  =  ? 

9-7  =  ? 

19-7=? 

1  +  3=? 

11  +  3  =  ? 

9-3  =  ? 

19-3=? 

4  +  5  =  ? 

14  +  5  =  ? 

G-B  =  ? 

16-5  =  ? 

6  +  3  =  ? 

16  +  3  =  ? 

6-2  =  ? 

16-2  =  ? 

1  +  7  =  ? 

11  +  7  =  ? 

3-2  =  ? 

13-2  =  ? 

1  +  4  =  ? 

11  +  4  =  ? 

4-1  =  ? 

14-1  =  ? 

8  +  3  =  ? 

11  +  6  =  ? 

9-6=? 

13-5  =  ? 

6  +  1=? 

13  +  3  =  ? 

7-2=? 

14-4  =  ? 

8  +  5  =  ? 

12  +  5  =  '^ 

8-3  =  ? 

15-6  =  ? 

7  +  7  =  ? 

15  +  2  =  ? 

9-5  =  ? 

18-9  =  ? 

54  ELEMENTS   OF   ARITHMETIC, 


31.  Take  ten  splints  and  tie  them  together.  Take 
ten  more,  and  make  another  bundle.  How  many  tens 
have  you? 

Two  tens  are  called  tiventy.     They  are  written  ^0, 

Take  another  bundle  of  ten.  How  many  tens  have 
you  now  ? 

Three  tens,  called  thirty^  are  written  30 ;  four  tens, 
forty ^  are  written  ^0  ;  five  tens,  fifty ^  are  written  60  ; 
six  tens,  %ixty^  are  written  60  ;  seven  tens,  seventy^  are 
written  /O ;  eight  tens,  eighty^  are  written  SO ;  nine 
tens,  ninety^  are  written  ^0. 

Count  by  tens  to  ninety. 

Write  the  numbers  from  ten  to  ninety  on  your  slate. 

Begin  at  90  and  count  back  by  tens  to  0. 

Put  away  all  but  two  bundles  of  tens.  Ten  and  ten 
are  how  many  ? 

How  many  fives  are  there  in  10  ?     How  many  in  20  ? 

How  many  twos  are  there  in  10?     How  many  in  20? 

Count  by  2's  to  20,  writing  the  numbers  on  your  slate, 
thus:  0,  2,  ^,  etc. 

Count  by  4's  in  the  same  way.  Count  by  5's  in  the 
same  way. 

A  boy  had  20  quarts  of  berries  and  sold  one  half  of 
them.    How  many  did  he  sell?    How  many  had  he  left? 

There  were  15  birds  on  a  tree.  How  many  more 
must  come  to  make  20  ? 

How  many  dimes  can  you  get  for  20  cents  ? 

How  many  half  dimes  can  you  get  for  20  cents  ? 


TWENTY  —  20.  55 

There  are  two  pints  in  a  quart.  How  many  quarts 
are  there  in  20  pints? 

Four  quarts  make  a  gallon.  How  many  gallons  are 
there  in  20  quarts  ? 


SLATE 

WORK. 

10- 

-7=? 

10-6  =  ? 

lof  10  =  ? 

1x2  =  ? 

20- 

-7  =  ? 

20-6  =  ? 

lof  20  =  ? 

2x2  =  ? 

10- 

-4  =  ? 

10-5  =  ? 

-iof  10  =  ? 

3x2  =  ? 

20- 

-4  =  ? 

20-5  =  ? 

lof20  =  ^ 

4x2  =  ? 

10- 

-2  =  ? 

20-5  =  ? 

10-   1  =  ? 

5x2  =  ? 

20- 

-2  =  ? 

20-2  =  ? 

20-   1  =  ? 

6x2=? 

10- 

-8  =  ? 

1x5=? 

20-  4  =  ? 

7x2  =  ? 

20- 

-8  =  ? 

2  5's  =  ? 

20-10  =  ? 

8x2  =  ? 

10- 

-3=? 

3  5\s  =  ? 

10-f-   4  =  ? 

9x2  =  ? 

20- 

~3  =  ? 

4  5's  =  ? 

10-10  =  ? 

10x2  =  ? 

11_4+   6  =  ?  20-7  +  6  =  ?  13-9  +  4  =  ?  20-8-4  =  ? 

11_8+  5  =  ?  20-2-9  =  ?  20-9-4  =  ?  16-7-8  =  ? 

17_8+  6  =  ?  20-7  +  6  =  ?  11-6  +  8  =  ?  19-2-8  =  ? 

12-6+  7  =  ?  12-7  +  6  =  ?  20-5-9  =  ?  20-7-5  =  ? 

15-6+  7  =  ?  14-7  +  9  =  '?  20-9-8  =  '?  19-8-7  =  ? 

20-6+  5  =  ?  20-9  +  4  =  ?  13-5  +  9  =  ?  18-8-6=? 

11_1+   8  =  ?  20-6-8  =  ?  18-9  +  3  =  ?  15-9-3=? 

17_9  +  10  =  ?  19-2  +  2  =  ?  20-8-7  =  ?  20-3-8  =  ? 

14_8+   7  =  ?  13-5  +  8=?  17-9  +  3  =  ?  18-8-6  =  ? 

13_94_11:=?  20-9-2  =  ?  20-4-8  =  ?  13-7-2  =  ? 

15_6  +  10  =  ?  18-8  +  6  =  ?  20-7-3  =  ?  19-8  +  2  =  ? 

19_5+  6  =  ^  20-6+4  =  -:>  20-8-6  =  ?  18-8-5  =  ? 

20-8-   3  =  ?  19-7-2  =  ?  15  +  4-7  =  ?  12  +  8-5=? 


56 


ELEMENTS   OF  ARITHMETIC. 


TENS  AND  ONES. 


□  □□iniaaai  aniLlLuLQQQDLaL  rn ni ni  n run m 

32.  Look  at  the  first  picture.  How  many  piles  of 
ten  blocks  do  you  see  ?     How  many  other  blocks  ? 

This  number  is  called  twenty-one^  and  it  is  written  ^/, 

How  many  tens  do  you  see  in  the  second  picture? 
How  many  ones  ? 

This  number  is  called  thirty-three^  and  it  is  written  33, 

How  many  tens  are  there  in  the  next  picture  ?  How 
many  ones  ? 

This  number  is  cdXl^di  forty-jive^  and  it  is  written  ^5. 

How  many  tens  are  there  in  the  last  picture  ?  How 
many  ones  ? 

This  number  is  called  jifty-ttvo^  and  is  written  6^. 

We  have  learned  to  read  tens  by  calling  a  number 
expressing  3  tens  thirty^  four  tens/or^?/,  etc. ;  and  when 
a  number  expresses  tens  and  ones,  we  read  it  by  naming 
the  tens  first  and  then  the  ones^  without  any  words 
between  them. 

We  can  tell  whether  a  figure  stands  for  ones  or  for 
tens  by  the  place  it  occupies.  If  it  represents  ones^  it 
has  the  first  place  to  the  rights  if  te7is  the  next  to  the  left. 

Read  the  following  numbers  by  telling  how  many  tens 
and  how  many  ones  there  are. 

Thus :  25  is  2  tens  and  5  ones,  or  twenty-five. 
37     14     83     96     12     20     21     38     82     29     30     26 
99     31     24     76     33     58     16     23     25     67     52     95 
68     77     53     74     83     76     39     26     34    51     90     50 


TWENTY-ONE  —  21.  57 

Begin  with  one  and  add  two  each  time  until  you  reach 
twenty-one. 

Count  by  threes  to  21. 

Three  feet  make  1  yard.  If  John  has  21  feet  of 
string  for  his  kite,  how  many  yards  has  he  ? 

How  many  weeks  are  there  in  21  days  ? 

If  Kate  makes  7  loaves  of  cake  every  day  for  3  days, 
and  sells  on  one  day  8  loaves  and  on  another  day  6,  how 
many  has  she  left? 

What  is  i  of  21  ?     What  is  |  of  21  ? 

If  Fred  buys  15  papers  of  one  kind  and  6  of  another, 
and  sells  8  of  them,  how  many  has  he  left  ? 

How  much  must  Jamie  pay  for  3  balls,  if  their  prices 
are  6  cents,  9  cents,  and  4  cents?  How  many  cents 
will  he  have  left,  if  he  had  21  cents  at  first  ? 

Harold  worked  10  examples  one  day,  7  the  next  day, 
and  4  the  next.  How  many  did  he  work  in  the  three  days  ? 
If  all  but  four  were  correct,  how  many  were  correct  ? 

If  I  have  21  nuts,  to  how  many  children  can  I  give  7 
each  ? 

If  I  rode  north  6  miles,  then  east  11  miles,  then  south 
6  miles,  how  far  was  I  from  the  starting-place  ? 

If  I  traveled  from  home  11  miles  on  Monda}^  7  miles 
farther  on  Tuesday,  and  3  miles  on  Wednesday ;  and 
on  my  way  back  on  Thursday  walked  10  miles  and  on 
Friday  6  miles,  how  many  miles  was  I  then  from  home  ? 


9  +  2  =  ? 

49  +  2  =  ? 

21-7  =  ? 

3x3  =  ? 

7  +  4  =  ? 

37  +  4  =  ? 

iof21=^ 

4x3=? 

8  +  3  =  ? 

78  +  3  =  ? 

|of  21  =  ^ 

5x3  =  ? 

19  +  2=? 

79  +  2  =  ? 

lof  21  =  ^ 

6x3=? 

17  +  4  =  ? 

47  +  4  =  ? 

f  of  21  =  ? 

7x3  =  ^ 

18  +  3  =  ? 

58  +  3=? 

1  of  21  =  ? 

1x7  =  ? 

58  ELEMENTS   OF   ARITHMETIC. 

RAPID   WORK. 

8-1-3  +  5-7-3  +  10-8  +  6-9  +  1  +  8  =  ? 

13-6  +  7-5-2  +  7  +  3-10-6  +  8  +  7-8  +  9=? 

11  +  8-7-6  +  13-10-7  +  6  +  8-9-3  +  8  +  9  =  ? 

21-2-7-4-5+6+8-9+4+9-6+7-4=? 

13  +  5  +  2-17  +  6  +  5-7  +  9-11  +  15-13  +  9-8  +  4  =  ? 

15-6  +  7-9  +  12-11  +  6-9-2  +  16-13  +  15-10  =  ? 

7  +  8-9  +  12-11  +  6-9-2  +  16-13  +  15-18  +  1  =  ? 


33.   Twenty  and  two  are  how  many  ? 

Count  by  twos  to  twenty-two. 

What  is  one  half  of  twenty-two  ? 

How  many  figures  are  required  to  write  twenty-two  ? 

For  what  does  the  left  hand  figure  stand  ? 

For  what  does  the  right  hand  figure  stand  ? 

Donald  has  3  pansies,  4  pinks,  5  violets,  6  roses,  and 
4  lilies.     How  many  flowers  has  he  ? 

There  are  22  children  in  Fred's  class.  If  11  stand  in 
a  row,  how  many  rows  are  there  ? 

22-2-2-2,  etc.,  to  0  18  +  ?=22  20  +  ?  =  22 

22-3-3-3,  etc.,  to  1  13  +  ?  =  22  17+?  =  22 

22-4-4-4,  etc.,  to  2  15+?  =  22  14  +  ?=22 

22-5-5-5,  etc.,  to  2  19  +  ?  =  22  16+?=22 


TWENTY-TWO  —  22.  59 

How  many  two-cent  postage  stamps  can  I  buy  for  22 
cents  ?     For  20  cents  ?     For  10  cents  ?     For  18  cents  ? 

1  17 

2  15  This  is  the  way  Emma  adds  a  column  and  tests  it  to 

3  12       see  if  it  is  correct. 

2^  ^\  She  found  the  sum  to  be  18.     She  then  proved  her 

o  q  result  correct   by  taking  in  succession   each  number 

0  n  from  18,  thus:  1  from  18  leaves  17  ;  2  from  17  leaves 
^  ^  15  ;  3  from  15  leaves  12  ;  1  from  12  leaves  11  ;  2  from 

1  ^11  leaves  9,  etc.    After  taking  away  the  last  number, 

2  3  0  was  left.     Test  your  results  in  the  same  way  in  the 

3  0  following  examples. 

18 

Add  and  test : 

8231  30  20  3321 

2332  20  30  4323 

2         2         2         3  10  30  3         3         2         3 

2  1  3  2  30  20  4  3  2  3 
3312  20  10  3323 
2233  10  20  4323 
2223  30  30  3323 
1233  20  30  4323 

3  4         1         3  10  20  3         3         2         3 


The  sign  $  stands  for  dollars. 

14  IT  $5  6  8  5  9  4  3  2  1 
42534237136 
47565426442 
42533633858 
42021311243 
12121111141 


22-i-2  =  ?         22^11  =  ?         1  of  22  =  ?       J- of  22  =  ? 
^\of22  =  ?       -j\of22  =  ?       j^of22  =  ?       ^^^  of  22  =  ? 


60  ELEMENTS  OF   ARITHMETIC. 

34.  20  boys  were  snow-balling  and  3  more  joined 
tliem.     How  many  were  there  then  ? 

If  each  boy  threw  1  snow-ball,  how  many  snow-balls 
were  thrown? 

If  all  but  3  hit  the  mark,  how  many  snow-balls  hit  the 
mark  ? 


9+   4  =  ? 

10  +  4  =  ? 

29  +  4  =  ? 

39  +  4=? 

6+   T  =  ? 

16  +  7  =  ? 

26  +  7=? 

36  +  7  =  ? 

8+  .5  =  ? 

18  +  5  =  ? 

48  +  5=? 

78  +  5=? 

3  +  10  =  ? 

19  +  4  =  ? 

49  +  4  =  ? 

72  +  9  =  ? 

Write  the  numbers  from  20  to  30;  from  30  to  40; 
from  19  to  10;  from  50  to  60;  from  50  to  40;  from  80 

to  90. 

What  is  the  largest  number  you  can  write,  using  but 
one  figure  ?     What  is  the  smallest  ? 

What  is  the  smallest  number  you  can  write,  using 
only  two  figures? 

What  is  the  largest  number  that  can  be  written  with 
two  figures  ? 

Add  first  by  lines  and  then  by  columns : 
2+5+3+4+1+2+3+3=? 
6+4+3+3+1+3+2+3=? 
5+5+5+5+2+2+2+2=? 
4+4+4+4+4+1+1+0=? 
1+2+4+3+4+3+4+3=? 


Add,  and  test  your  results 

as 

before 

;: 

2       3       3 

2 

4 

4 

6 

Q 

2 

3 

2 

4 

2       2       4 

9 

3 

2 

7 

9 

4 

4 

6 

4 

8       13 

1 

2 

5 

2 

5 

6 

3 

3 

4 

4       9       5 

4 

1 

3 

3 

3 

7 

4 

2 

4 

7       6       5 

6 

If 
7 

4 

5 

3 

4 

3 

6 

6 

'/^^' 


TWENTY-FOUR  — 24.  "  ^-^Gl 

If  23  passengers  travel  by  coach  and  7  are  seated  on 
top,  liow  many  are  there  inside  ? 

Morton  earned  15  cents  and  his  uncle  gave  him  8 
cents  more.  He  then  bought  3  oranges  at  3  cents  apiece. 
How  many  cents  had  he  left  ? 

If  3'OU  have  13  cents,  how  many  more  cents  do  you 
need  to  buy  a  book  that  costs  20  cents  ? 

How  many  more  cents  do  you  need  to  buy  one  that 
costs  23  cents? 

Twenty-three  children  belong  to  Miss  C's  class,  but 
only  19  are  present.     How  many  are  absent  ? 

There. were  6  persons  in  a  car,  when  7  more  were 
taken  in;  soon  5  left  and  9  got  on.  How  many  were 
there  then  in  the  car  ? 

If  the  car  could  seat  23  people,  how  many  more  could 
be  seated  ? 

If  a  lady  goes  to  Boston  with  23  dollars  and  spends 
16  dollars,  how  many  dollars  has  she  left  ? 

35.    Count  by  2's  to  24. 

Count  by  3's  to  24,  then  back  to  0. 

Count  by  4's  to  24,  then  back  to  0. 

How  many  two-cent  postage  stamps  can  I  buy  for  24 
cents  ? 

How  many  yards  are  there  in  24  feet  ? 

How  many  gallons  are  there  in  24  quarts  ? 

Twelve  things  make  a  dozen.  How  many  dozen  but- 
tons has  Ethel,  if  she  has  24  buttons  ? 

Mr.  B.  had  24  eggs.  He  put  them  into  boxes  holding 
6  eggs  each.     How  many  boxes  did  he  fill  ? 

If  the  boxes  had  held  12  eggs,  how  many  boxes  would 
he  have  used? 


62 


ELEMENTS   OF   ARITHMETIC. 


Write  all  the  pairs  of  numbers,  which,  added  together, 

make  24. 

24-2-3-4-4-3-6=? 

24-2-2-2,  etc.,  to  0. 

24-3-3-3,  etc.,  to  0. 

Subtract  each  number  less  than  10  from  24  as  many 
times  as  you  can,  and  write  the  results  on  your  slate. 


1x4  =  ? 

3x3  =  ? 

12- 

-6  =  ? 

24- 

-  4  =  ? 

2x4  =  ? 

4x3  =  ? 

12- 

-2  =  '^ 

24- 

-   8  =  ? 

3x4  =  ? 

5x3  =  ? 

9- 

-3  =  ? 

24- 

-12  =  ? 

4x4  =  ? 

6x3  =  ? 

18- 

-3  =  ? 

24- 

_   2  =  ? 

5x4  =  ? 

7x3  =  ? 

12- 

-4  =  ? 

1  of  24  =  ? 

6x4  =  ? 

8x3  =  ? 

16- 

-4  =  ? 

iof24=? 

1x6  =  ? 

1x8  =  ? 

24- 

-3  =  ? 

fof  24  =  ? 

2x6  =  ? 

2x8  =  ? 

21- 

=-3  =  ? 

4  of  24  =  ? 

3x6  =  ? 

3x8  =  ? 

8- 

-4  =  ? 

fof  24  =  ? 

4x6  =  ? 

24-3  =  ? 

8- 

-2  =  ? 

1  of  24  =  ? 

1x3  =  ? 

18-2  =  ? 

15- 

-5  =  ? 

fof  24=? 

2x3=? 

24-6  =  ? 

12- 

-3  =  ? 

iof24  =  ? 

Twelve  inches  make  a  foot.  How  many  feet  are 
there  in  24  inches  ? 

Harold  has  2  dimes  and  4  cents.  How  much  money 
has  he  ?  How  many  bananas,  at  6  cents  each,  can  he 
buy  with  his  money  ? 

If  he  should  prefer  to  buy  oranges  at  3  cents,  how 
many  could  he  buy  ? 

If  he  should  give  half  his  money  to  his  brother,  how 
much  would  each  have  ? 

David  walked  24  miles  in  8  hours.  How  far  did  he 
walk  in  one  hour? 

KachBl  had  24  postal  cards.     She  used  6  at  one  time, 


TWENTY-FIVE  —  25.  63 

5  at  another  time,  and  7  at  another  time.  How  many 
had  she  left  ? 

Daisy  had  2  dozen  dolls'  cups.  She  piled  them  up, 
placing  half  a  dozen  in  a  pile.     How  many  piles  had  she  ? 

Harold,  Reuben,  James,  and  Burton  have  6  cents  each, 
and  each  one  buys  a  peach  for  2  cents.  How  much 
more  money  do  they  need  to  buy  a  ball  for  20  cents? 

A  piece  of  cloth  contained  24  yards ;  1  of  it  was  sold. 
How  many  yards  were  sold  ?  How  many  yards  remained 
unsold  ? 

Thornton  bought  2  dozen  postal  cards.  How  much 
did  he  pay  for  them  ?  If  he  gave  the  clerk  25  cents,  how 
much  change  should  he  receive  ? 

Mr.  B,  worked  at  his  shop  from  7  o'clock  in  the  morn- 
ing until  12  o'clock  noon,  and  from  1  to  4  o'clock  in 
the  afternoon,  and  at  home  he  worked  from  7  to  9 
o'clock.     How  many  hours  did  he  work  ? 

36.  Lucy  has  two  dimes  and  a  half-dime.  To  how 
many  cents  are  they  equal  ? 

She  spent  all  but  4  cents  for  3  paper  dolls.  How 
many  cents  did  she  spend  ?  What  was  the  cost  of  each 
doll? 

If  Lulu  paid  25  cents  for  a  slate  and  sponge,  and  the 
sponge  alone  cost  6  cents,  what  was  the  price  of  the 
slate  ? 

A  milkman  who  had  25  quarts  of  milk  put  it  in  gal- 
lon (4  quart)  cans.  How  many  cans  did  he  fill,  and 
how  much  milk  had  he  left  ? 

How  many  people  could  he  supply  with  1  quart  a 
day  ?  With  5  quarts  ?  With  3  quarts,  and  how  much 
would  be  left  ? 


64  ELEMENTS   OF   ARITHMETIC. 

Harold  is  9  years  old,  and  Arthur  is  4  years  younger. 
In  how  many  years  will  Arthur  be  25  years  old  ?  How 
many  years  before  Harold  will  be  25  ? 

Daisy  had  25  acorns,  which  she  placed  in  5  equal 
piles.     How  many  did  she  put  in  each  pile  ? 

What  is  1  of  25  cents?  -J-  of  25  children?  -J-  of  25 
hens  ?     I"  of  25  trees  ? 

Lena  had  a  quarter  of  a  dollar  (25  cents),  and  spent 
8  cents.     How  much  had  she  left  ? 

The  cook  used  2  dozen  eggs  for  the  Christmas  dinner. 
If  she  had  25  eggs  at  first,  how  many  had  she  left  ? 

How  many  half-dimes  can  you  get  in  change  for  a 
quarter  of  a  dollar  ? 

24  +  1  =  ?  25-4-4-4,  etc. 

25-2-2,  etc.  25-5-5-5,  etc. 

25-3-3-3,  etc.  25-6-6-6,  etc. 

1x5  =  ?  6x   5  =  ?  25-f-  1  =  ?  20+?  =  25  ?-fl6  =  25 

2x5  =  ?  Tx   5  =  ?  iof25  =  ?  15+?  =  25  ?  +  14  =  25 

3x5  =  ?  8x   5  =  ?  |of25  =  ?  23  +  ?  =  25  ?  +  21  =  25 

4x5  =  ?  25-  5  =  ?  f  of25  =  ?  5  +  ?  =  25  ?  +  ll  =  25 

5x5  =  ?  25-10  =  ?  |of25  =  '>  7-|-?=25  ?-M8  =  25 

37.    Two  tens  and  six  are  how  many? 

If  I  have  26  pictures  and  give  away  one  half  of  them, 
how  many  do  I  give  away?     How  many  have  I  left? 
2x?  =  26  26-13  =  ? 

Write  all  the  pairs  of  numbers  you  can  whose  sum 
equals  26. 

Agnes  bought  26  buttons  at  the  store.  The  buttons 
were  on  cards,  having  12  buttons  each.  How  many 
cards  did  she  buy,  and  how  many  buttons  besides  ? 


TWENTY-SIX  — 26.  ^^ 

In  our  school  there  are  23  pupils.  On  Saturday  5 
boys  and  3  girls  went  skating,  6  girls  and  4  boys  went 
coasting,  and  the  rest  went  to  Boston.  How  many  pupils 
went  to  Boston  ? 

James  solved  7  examples  on  Monday,  8  on  Tuesday, 
and  6  on  Wednesday.     How  many  did  he  solve  in  all  ? 

A  pole  stands  7  feet  in  the  mud,  6  feet  in  the  water, 
and  9  feet  above  water.     How  long  is  the  pole  ? 

I  counted  the  birds  on  4  telegraph  wires.  On  the  first 
wire  there  were  6  birds,  on  the  second,  5,  on  the  third, 
6,  and  on  the  fourth,  6.     How  many  birds  did  I  count? 

A  boy  had  6  copper  cents,  2  half-dimes,  and  a  dime. 
How  much  money  had  he  ? 

Ethel  has  five  half-dimes  and  a  cent.  How  much 
money  has  she  ? 

How  many  Christmas  cards,  at  4  cents  each,  can  Ethel 
buy  with  her  money,  and  how  many  cents  will  she  have 
left? 

Frank  is  26  years  old,  and  Philip  is  19.  What  is  the 
difference  in  their  ages  ? 

A  newsboy  buys  16  Heralds  and  10  Transcripts.  He 
sells  12  Heralds  and  7  Transcripts.  How  many  papers 
has  he  yet  on  hand  ? 

A  man  had  26  gallons  of  vinegar.  He  first  sold  one 
half  of  it,  and  later  he  sold  4  gallons  more.  How  many 
gallons  did  he  sell  ?     How  many  had  he  left  ? 

A  boy  had  12  slate  pencils  and  11  lead  pencils ;  he 
gave  away  half  the  slate  pencils.  How  many  pencils 
had  he  left? 

Count  by  2's  to  26. 

Write  the  table  of  2's  to  26,  thus : 

1x2=?  2x2  =  ?  3x2  =  ?  etc. 

EL.    OF   AR.  —  5 


ee 


ELEMENTS   OF   AKITHMETIC. 


38.   If  there  are  10  marbles  in  each  of  the  bags  in 

the  picture,  how  many  mar- 
bles are  there  in  all? 

Two  tens  and  7  ones  are 
how  many? 

Two  dimes  and  7  cents  are 
how  many  cents  ? 

Fred  had  26  apples  in  his 
basket,  and  1  in  his  hand. 
How  many  had  he  ? 
To  how  many  children  can  he  give  3  apples  each? 
9  each  ?     5  each,  and  how  many  will  be  left  ? 
How  many  yards  are  there  in  27  feet? 
If  I  divide  27  pieces  of  candy  equally  among  3  chil= 
dren,  how  many  pieces  will  each  have  ? 

What  is  one  third  of  27  ?     Two  thirds  of  27  ? 
A  grocer  having  27  barrels  of  flour,  put  them  in  9 
equal  piles.    How  many  barrels  were  there  in  each  pile  ? 
What  is  one  ninth  of  27  melons  ?      Two  ninths  ? 
If  I  have  f  of  27  dollars,  how  many  dollars  have  I  ? 
A  man  having  27  cows  sold  |  of  them.     How  many 
did  he  sell  ?     How  many  had  he  left  ? 

A  fruit  dealer  found  that  ^  of  his  27  baskets  of  peaches 


was  spoiled.     How  many  were   spoiled? 
baskets  of  good  peaches  had  he  left  ? 


4x3  =  ? 
1x3  =  ? 
6x3  =  ? 
2x3  =  ? 
8x3  =  ? 
9x3  =  ? 


3x3  =  ? 
7x3  =  ? 
5x3  =  ? 
1x9  =  ? 
2x9  =  ? 
3x9  =  ? 


4x7  =  ? 
27-^3  =  ? 

27-^9  =  ? 

18-3  =  ? 

12-3  =  ? 

3)3( 


6-3  =  ? 
24-3  =  ? 

24-8  =  ? 

18-3  =  ? 
3)9( 
7)21( 


How   many 

lof  27  =  ? 
iof  21  =  ? 
lof  15  =  ? 

n 


J  of  27 


18^2  =  ? 
18-9  =  ? 


TWENTY-SEVEN  —  27.  67 

Add  by  lines  and  columns : 

3+3+3+3+3+3+3+3=^  4+8+4+7+4+1=? 

4+2+5+7+1+6+0+2=?  5+4+5+3+4+6=? 

8+4+1+9+0+2+2+2=?  5+3+1+7+5+4=? 

6+6+3+4+4+2+1+5=?  7+2+8+6+3+3=? 

7+8+2+5+4+5+4+1=?  6+4+2+4+4+7=? 

27  +  1  =  ? 

Count  by  2's  to  28. 

Count  by  4's  to  28. 

Count  by  7's  to  28. 

How  many  tens  and  how  many  ones  are  there  in  28  ? 

If  Nellie  puts  28  peaches  on  four  plates,  putting  the 
same  number  on  each  plate,  how  many  will  there  be  on 
each  plate  ? 

If  she  gives  away  the  peaches  on  one  plate,  what  part 
of  the  whole  will  she  give  away  ?     What  is  ^  of  28  ? 

Max  bought  7  pairs  of  shoes  for  28  dollars.  What 
was  the  cost  of  1  pair  ? 

What  is  I  of  28?     fof28?     f  of  28? 

How  many  five-cent  pieces  and  how  many  cents  will 
make  28  cents  ? 

Fred  was  sick  for  28  days.  How  many  weeks  was 
that? 

Mro  Brown  had  28  pecks  of  potatoes,  which  he  sold 
by  the  bushel  (4  pecks) o  How  many  bushels  did  he 
sell? 

28  pints  equal  how  many  quarts  ? 
28  quarts  equal  how  many  gallons  ? 

William  spent  one  half  of  his  28  cents  for  a  slate  and 
5  cents  for  a  lead  penciL  How  much  did  he  spend? 
How  much  had  he  left  ? 


68  ELEMENTS  OF  ARITHMETIC. 

28-5-9-7-3-4=? 
28-3-6-2-2-1=? 
28-=9-2-8-l-7  =  ? 
28-4-4-4-4-4  =  ? 

28-T-T-T-T=? 

Add  and  prove  by  subtracting  as  on  page  59. 


4 

5 

6 

2 

1 

3 

4 

5 

6 

7 

3 

2 

4 

4 

6 

2 

8 

4 

3 

3 

4 

4 

9 

4 

9 

4 

4 

4 

7 

4 

0 

7 

5 

3 

4 

2 

1 

3 

4 

6 

4 

4 

28  +  1  =  '? 

Beginning  with  1  count  by  2's  to  29, 
Beginning  with  1  count  by  4's  to  29. 
Beginning  with  1  count  by  7's  to  29, 

39.  After  giving  9  cents  for  some  candy,  Anita  had 
2  dimes  remaining«  How  many  cents  did  she  have  at 
first? 

At  9  cents  a  spool,  how  much  will  3  spools  of  silk 
cost  ? 

One  day  Alice  worked  4  hours  at  7  cents  an  hour, 
and  gave  10  cents  for  car  fare.  How  many  cents  had 
she  left? 

Henry  had  29  cents,  but  spent  10  cents  for  a  top. 
How  much  had  he  left?  How  much  did  he  lack  of 
having  a  quarter  of  a  dollar? 

A  boy  who  had  9  cents  went  on  3  errands,  earning  5 
cents  for  each.     How  much  money  did  he  have  then  ? 

An  organ  grinder  received  at  one  house  10  cents,  at 


TWENTY-NINE  —  29.  69 

another  10  cents,  and  at  another  9  cents.     How  much 
did  he  receive  from  them  all  ? 

Henry  caught  15  fishes  and  Oren  14.  How  many  did 
both  catch  ? 

After  spending  19  cents  for  fishing  tackle,  a  boy  had 
10  cents  left.     How  much  had  he  at  first  ? 

Mary  went  to  school  18  days  in  January  and  11  in 
February.  How  many  days  did  she  attend  in  those 
months  ? 

29-5-4-3-6=? 
29-7-3-4-2=? 
29-8-8-8-3  =  ? 
29-5-6-4-9=? 
29-8-3-9-9=? 

40.  Morton  paid  3  dimes  for  a  picture-book.  What 
did  it  cost  ? 

Lulu  had  3  dimes,  and  spent  one  half  her  money.  How 
many  cents  did  she  spend  ? 

Susan's  clothes-line  is  10  yards  long.  How  many 
feet  long  is  it  ? 

How  many  half-dimes  can  I  get  in  exchange  for  30 
cents  ? 

How  many  oranges,  at  5  cents  apiece,  can  be  bought 
for  3  dimes  ? 

Lena  bought  half  a  dozen  bananas  at  5  cents  each. 
What  did  they  cost  ? 

Lucy  has  25  cents  and  a  half-dime.  How  many  quarts 
of  cherries,  at  6  cents  a  quart,  can  she  buy  with  the 
money  ? 

Clarence  lives  6  blocks  from  the  schoolhouse.  How 
many  blocks  does  he  walk  to  and  from  school  each  day, 
if  he  goes  home  to  dinner  ? 


70  ELEMENTS  OF   ARITHMETIC. 

Some  girls  wanted  to  buy  a  game  that  cost  30  cents ; 
3  gave  2  cents  each ;  4  gave  3  cents  each ;  2  gave  4 
cents  each.  How  many  girls  contributed?  How  much 
more  do  they  need  to  pay  for  the  game  ? 

Leila  bought  3  pounds  of  sugar  at  7  cents  a  pound, 
and  gave  the  storekeeper  3  dimes.  What  change  should 
she  receive  ? 

10  +  ?  =  30  18  +  ?  =  30  14  +  ?  =  30 

17  +  ?  =  30  15-f?  =  30  12  +  ?  =  30 

20  +  ?-30  19  +  ?  =  30  ll  +  ?  =  30 

25  +  ?  =  30  16  +  ?  =  30  30-8  =  ? 

From  30  take  2  as  many  times  as  you  can,  naming 
results ;  as,  28,  26,  etc. 

In  the  same  way  subtract  3,  5,  6,  7,  4,  8. 

Anson  had  30  rabbits,  but  10  of  them  got  away.  How 
many  had  he  left  ? 

Mary  had  2  dimes  and  a  five-cent  piece.  After  pay- 
ing 10  cents  for  a  composition  book  and  5  cents  for  a 
pencil,  how  much  had  she  left  ? 

Hannah  attended  school  every  school  day  for  30  days, 
but  her  sister  Sarah  was  absent  8  days  during  this  time. 
How  many  days  did  Sarah  attend  ? 

A  month  is  commonly  regarded  as  30  days.  How 
many  days  is  half  a  month  ? 

The  fare  from  Andes  to  Ray  field  is  15  cents.  A 
pupil's  ticket  from  Andes  to  Rayfield  and  back  is  only 
18  cents.  How  much  cheaper  is  the  pupil's  ticket  than 
the  regular  fare  ? 

A  boy  who  had  3  dimes  paid  25  cents  for  a  ticket  to 
a  museum.     How  much  money  had  he  left? 


REVIEW   EXERCISES.  71 


REVIEW   EXERCISES. 
41.   Answers  to  be  given  instantljo 


iof  10  =  ? 

16-9  =  ? 

20-5  =  ? 

19-7=? 

lof  20  =  ? 

15-3  =  ? 

24-4  =  ? 

3  3's=? 

7x2  =  ? 

15-8  =  ? 

30-10  =  ? 

12-4  =  ? 

30-5  =  ? 

25-8  =  ? 

6  +  7=? 

18-6  =  ? 

9  +  5  =  ? 

3  10's  =  ? 

11-9  =  ? 

2x8  =  ? 

iof2T  =  ? 

4)28( 

4  +  7=? 

12  2's  =  ? 

|of27  =  ^ 

28-7  =  ? 

8-3  =  ? 

iof  30=? 

30-3  =  ? 

13-9  =  ? 

18-3  =  ? 

8  2's=? 

28^7  =  ^ 

ii-2=:> 

3x7=? 

21-3  =  ? 

.iof28  =  ^ 

9  +  9  =  ? 

13-7  =  ? 

7x2  =  ? 

13-8  =  ? 

19  +  9  =  ? 

9  +  8  =  ? 

6x5  =  ? 

3x9  =  ^ 

11-6  =  ? 

19+-8  =  ? 

4x6  =  ? 

9x3  =  ? 

5  +  7  =  ? 

5x5=? 

24-3  =  ? 

12-5  =  ? 

17-8  =  ? 

iof25='> 

lof  16  =  ? 

lof24  =  ^ 

3x6  =  ^ 

lVof20  =  '? 

iofl8  =  ? 

f  of  24  =  ? 

9x3  =  ? 

{of  21  =  ? 

J^of20  =  ? 

24-12  =  ? 

7x3  =  ? 

12  +  6  =  ? 

4x8  =  ? 

24-8  =  ? 

18-3  =  ? 

24-6  =  ? 

8x2=? 

9-5  =  ? 

8  3's  =  ? 

3x5  =  ? 

22-2  =  ? 

3x4  =  ? 

16-5  =  ? 

lof21=? 

7x3=? 

3x8  =  ? 

12-8  =  ? 

f  of  21=? 

21-7  =  ? 

lof  16  =  '^ 

7  +  8  =  ? 

13-4  =  ? 

iof  16=? 

1  of  16  =  ? 

4  3's  =  ? 

iof  24  =  ? 

f  of  16  =  ? 

2x12  =  ? 

14-9  =  ? 

5  3's  =  ? 

6  3's=? 

9-5  =  '^ 

19-3  =  ? 

15-8  =  ? 

8  +  7=? 

72  ELEMENTS  OF   ARITHMETIC. 


19-5  =  ? 

25-5  =  ? 

4  4's  =  ? 

2  +  9  =  ? 

29-5  =  ? 

7  4's  =  ? 

6  4's  =  ? 

11  2\s  =  ? 

8)24( 

12-9  =  ? 

3  6's  =  ? 

4x7  =  ? 

11  2's  =  ? 

17-8  =  ? 

lof24  =  ? 

iof24  =  ? 

2x9  =  ? 

30-5  =  ? 

5  +  9  =  ? 

f  of  24  =  ? 

3x9  =  ? 

23-7  =  ? 

3  +  5  =  ? 

18-2  =  ? 

9)27( 

3x6  =  ? 

13  +  5  =  ? 

-1  of  30  =  ? 

18-9  =  ? 

19-8  =  ? 

28-f-4  =  ? 

2x9  =  ? 

18-9  =  ? 

29-8  =  ? 

8  3's  =  ? 

22  —  11  =  ? 

5  6's  =  ? 

20-4  =  ? 

2  8's  =  ? 

iof  9  =  ? 

NUMBERS   FROM    30   TO    100. 

42.   Count  by  4's  to  32. 
Write  the  table  of  4's,  thus ; 

1x4  =  ?  2x4  =  ?  3x4=?  etc. 

There  are  4  quarts  in  a  gallon.  How  many  quarts 
are  there  in  2  gallons  ?  In  4  gallons  ?  In  7  gallons  ? 
In  8  gallons  ? 

There  were  4  sleds  on  Morton's  hill  last  Saturday. 
If  each  one  held  8  children,  how  many  children  were 
coasting  on  the  hill  ? 

1x8  =  ?  2x8  =  ?         3x8  =  ?  4x8=-:^ 

Bertie  had  32  cents.  If  he  gave  Helen  ^  of  his 
money,  how  many  cents  did  he  give  her  ? 

Mr.  Young  had  32  sheep,  but  he  sold  |  of  them. 
How  many  did  he  sell?     How  many  had  he  left? 

John  had  30  cents ;  he  spent  one  half  of  it  for  a  slate, 
and  2  cents  for  a  sponge.  How  much  had  he  left? 
Then,  if  his  mother  gave  him  10  cents,  how  much  had  he? 


NUMBERS  FEOM  30  TO   100.  73 

How  many  pecks  (8  quarts)  are  there  in  32  quarts  ? 
How  many  3-cent  pieces  can  I  get  for  3  dimes  and  3 
cents  ? 

Write  the  table  of  3's,  thus : 

Ix   3  =  ?  2x    3  =  ?  3x   3  =  ?etc. 

1x11  =  ?  2x11  =  ?  3x11  =  ? 

Julia  had  33  cents,  and  Olive  J-  as  many.  How 
many  cents  had  Olive? 

33-3  =  ?      33-11  =  ?      iof33  =  ?      J^  of  33=? 

A  Avalk  measured  11  yards  in  length.  Since  a  yard 
is  3  feet  long,  how  many  feet  long  was  the  walk  ? 

I  spent  33  cents  for  balls  at  11  cents  apiece.  How 
many  balls  did  I  buy  ? 

Count  by  2's  to  34,  and  back  from  34  to  0. 

17  +  17=?  2x17  =  ?  34-2  =  ? 

Two  men  start  from  the  same  place  and  travel  in 
opposite  directions.  One  travels  at  the  rate  of  3  miles 
an  hour,  the  other  at  the  rate  of  2  miles  an  hour.  How 
far  apart  are  they  at  the  end  of  2  hours?  5  hours? 
7  hours  ?     8  hours  ?     6  hours  ? 

Count  by  5's  to  60.  Count  by  7's  to  70.  Count  by 
lO's  to  70. 

Donna  worked  7  examples  each  day  for  5  days.  How 
many  did  she  work  ? 

Mrs.  H.  bought  from  the  milkman  35  quarts  of  milk 
in  one  week.  What  was  the  average  number  of  quarts 
she  bought  per  day  ? 

How  many  days  are  there  in  2  weeks  ?  In  5  weeks  ? 
In  4  weeks  ?  In  3  weeks  ?  In  6  weeks  ?  In  8  weeks  ? 
In  7  weeks  ? 


74  ELEMENTS  OF  ARITHMETIC. 

If  5  men  can  do  a  piece  of  work  in  7  days,  how  long 
will  it  take  1  man  to  do  it  ? 

How  many  weeks  are  there  in  35  days  ?  In  14  days  ? 
In  28  days?     In  42  days? 

-iof35  =  ?  iof35  =  ?  35-7-? 

I  of  35  =  ?  fof35  =  ?  35-5  =  ? 

Two  men  start  from  the  same  place  and  travel  in 
opposite  directions.  One  travels  9  miles  an  hour,  the 
other  travels  7  miles  an  hour.  How  far  apart  will  they 
be  in  2  hours  ? 

Suppose  the  men  traveled  in  the  same  direction,  how 
far  apart  would  they  be  in  2  hours  ?  In  7  hours  ?  In 
11  hours?     In  12  hours? 

Harry  earned  7  half-dimes  while  Lucy  earned  3  dimes 
and  a  five-cent  piece.  Which  earned  the  larger  sum  of 
mone}^  ? 

At  7  cents  a  yard,  what  is  the  cost  of  4  yards  of 
cambric?  Of  3  yards?  Of  5  yards?  Of  7  yards? 
Of  6  yards  ? 

How  many  weeks  was  Lulu  away  at  the  seashore,  if 
she  was  gone  21  days  ?  14  days  ?  28  days  ?  42  days  ? 
63  days  ?     49  days  ?     b^  days  ? 

1x6=?  2x6=?  3x6=?  4x6=?  5x6=?  ^xQ>=l 
7x6  =  ?  8x6  =  ?   9x6  =  ?  10x6  =  ?  11x6  =  ?  12x6=? 

Emily  is  18  years  old,  and  her  uncle  is  twice  as  old. 
How  old  is  he  ? 

What  is  i-  of  1 36  ?     J  of  36  men  ?     -|  of  36  bushels  ? 

How  many  times  is  4  contained  in  12?  In  24?  In 
36?     In  20?     In  16?     In  8?     In  28  ?     In  32  ? 

There  are  6  working  days  in  a  week.  How  many 
working  days  are  there  in  2  weeks  ?     In  6  weeks  ?     In  3 


NUMBERS  FROM  30  TO   100.  75 


weeks  ?     In  5 

weeks  ?     In  4  weeks  ?     In 

7  weeks? 

In 

10  weeks?     I] 

:l   8 

weeks  ? 

In  9  weeks? 

In  11  weeks? 

In  12  weeks? 

12-  6  =  ? 

24- 

-6  =  ? 

30-6=? 

36-6=? 

24-  6  =  ^ 

m- 

-6  =  ? 

60^6  =  ? 

72-6  =  ? 

6x    6-'^x8 

6: 

k6=?x9 

6x6=?xl2 

5x6  =  ? 

x3 

2xl0  =  ?x4 

4; 

><4  =  2x? 

6x2=?x4 

4x4  =  ? 

x8 

48-  6  =  ? 

18- 

f-6  =  ? 

42-6  =  ? 

How  many  yards  of  cloth,  at  $3  a  yard,  will  pay  for 
5  barrels  of  flour  at  $  6  per  barrel  ? 

Willard  sold  12  barrels  of  apples  at  $8  per  barrel,  and 
spent  half  the  money  for  books  at  1 2  each.  How  many 
books  did  he  buy? 

Otto  sold  6  quarts  of  berries  at  6  cents  a  quart,  and 
spent  the  money  for  9  bananas.  How  much  did  he  pay 
apiece  for  the  bananas  ? 

There  are  9  children  in  the  drawing  class,  and  4  times 
as  many  in  the  arithmetic  class.  How  many  are  there 
in  the  arithmetic  class  ? 

At  4  cents  a  yard,  what  is  the  cost  of  2  yards  of 
ribbon?  Of  4  yards?  Of  6  yards?  Of  9  yards?  Of 
10  yards  ?     Of  8  yards  ?     Of  8  yards  ? 

How  many  weeks'  work  are  there  in  36  working 
days?  In  18?  In  24?  In  30?  In  12?  In  42?  In 
60?     In  72? 

Count  by  2's  to  38. 
9  +  9  =  ?       19  +  9  =  ?       29  +  9  =  ?  etc.        19  +  19  =  ?  etc. 

Count  by  3's  to  39.     Count  by  4's  to  48. 

At  4  cents  apiece,  what  must  I  pay  for  5  oranges? 
For  3?  For  8?  For  6?  For  10?  For  4?  For  7? 
For  9  ?     For  11  ?     For  1  dozen  ? 


76  ELEMENTS  OF   ARITHMETIC. 

Bert  rode  his  pony  8  miles  a  day  for  5  days.  How 
far  did  he  ride  ? 

How  many  eggs,  at  4  cents  each,  will  pay  for  5  yards 
of  ribbon  at  8  cents  a  yard  ? 

If  a  stage  goes  8  miles  an  hour,  how  many  hours  will 
it  take  to  travel  24  miles  ?  40  miles  ?  16  miles  ?  48 
miles  ?     32  miles  ?     56  miles  ?     72  miles  ?     64  miles  ? 

What  is  I  of  40?  i  of  40?  f  of  40?  ^  of  40? 
JoOf40?     3-3^  of  40?     iof40?     |of40?     |of40? 

How  many  live-cent  loaves  of  bread  can  Florence  buy 
with  25  cents  ?  With  35  cents  ?  With  40  cents  ?  With 
50  cents? 

How  many  gallons  are  there  in  12  quarts?  In  28 
quarts  ?  In  36  quarts  ?  In  40  quarts  ?  In  48  quarts  ? 
In  16  quarts? 

How  many  quarts  are  there  in  16  pints  ?  In  24  pints  ? 
In  20  pints  ?     In  8  pints  ?     In  40  pints  ?     In  38  pints  ? 

How  many  pecks  are  there  in  3  bushels?  In  8 
bushels?  In  10  bushels?  In  6  bushels?  In  4  bushels? 
In  9  bushels  ?     In  7  bushels  ? 

If  one  pencil  costs  5  cents,  what  is  the  cost  of  3  pen- 
cils? Of  5  pencils?  Of  4  pencils?  Of  2  pencils? 
Of  6  pencils?     Of  7  pencils? 

At  7  cents  a  quart,  what  is  the  cost  of  4  quarts  of 
blueberries  ?  Of  6  quarts  ?  Of  3  quarts  ?  Of  2  quarts  ? 
Of  7  quarts  ? 

Count  by  7's  to  70.     Count  by  8's  to  80. 

Lucy  bought  6  paper  dolls  at  7  cents  each.  She  gave 
the  clerk  50  cents.  What  change  should  she  receive? 
6x7=2x? 

Donna  has  9  half-dimes  in  her  pocket  book.  To  how 
many  cents  are  they  equal  ? 


NUMBERS   FROM  30  TO  100,  77 

How  many  toy  books,  at  6  cents  each,  can  she  buy,  and 
how  many  cents  will  she  have  left? 

If  Clarence  sleeps  8  hours  at  night,  how  many  hours 
does  he  sleep  in  3  nights  ?  In  5  nights  ?  In  7  nights  ? 
In  4  nights  ?     In  6  nights  ? 

Helen  had  3  five-cent  pieces,  and  Nannie  had  3  dimes. 
How  many  sheets  of  drawing  paper,  at  9  cents  each,  can 
they  buy  if  they  put  their  money  together  ? 

Count  by  5's  to  60.     Count  by  9's  to  90. 

How  many  5's  are  there  in  25  ?  In  40  ?  In  15  ?  In 
45?    In  30?     In  50?    In  35?    In  60?    In  20?    In  55? 

How  many  9's  are  there  in  18?  In  45?  In  27?  In 
36-?     In  54?  etc. 

If  a  boy  can  buy  9  marbles  for  a  cent,  how  many 
marbles  can  he  buy  for  3  cents  ?  For  2  cents  ?  For  5 
cents  ?     For  4  cents  ?     For  6  cents  ?     For  7  cents  ? 

Bert 's  pony  can  travel  8  miles  an  hour.  How  much 
farther  can  he  travel  in  6  hours  than  Harold's  pony, 
which  can  travel  only  6  miles  an  hour? 

(9x  5)-(5x6)  =  ?  (  4xl0)-(4x8)=? 

(4x  9)-(3x4)=?  (  5xl0)-(6x6)  =  ? 

(7x  7)-(6x7)-?  (  7x   4)-(9x3)  =  ? 

(6x  5)-(3x5)  =  ?  (  8x   6)-(5x4)=? 

(4xl0)-(5x7)  =  ?  (  5x   5)-(3x5)  =  ? 

(4x  4)  +  (5x2)=?  (12x   3)-(4x9)=? 

(6x  7)-(3x7)  =  ?  (  6x   4)  +  (3x5)=? 

(5x  8)-(3x3)=?  (  3x   6)  +  (3x4)=? 


(3xll)  +  (2x5)  =  ?         (  8x   4)-(4x7)  = 


How  many  lO's  are  there  in  30  ?     In  50  ?     In  90  ?     In 
100? 

What  is  I  of  50  cents  ?     Of  40  cents  ?     Of  30  cents  ? 


NOTATION    AND    NUMERATION. 


43.  We  have  learned  that  ten  ones  make  one  ten.  ISTow 
take  ten  tens  and  tie  them  together.  Ten  tens  are  called 
one  hundred. 

In  writing  hundreds,  the  figure  representing  hundreds  is 
placed  in  the  third  place  from  the  right. 
Thus,  one  liundred  is  written  100. 

44.  Write  with  figures  : 

Two  hundred.  Three  hundred.  Nine  hundred. 

Seven  hundred.  Four  hundred.  Six  hundred. 

Five  hundred.  Eight  hundred.  One  hundred. 

45.  When  numbers  are  expressed  by  three  figures,  the 
figure  at  the  right  represents  ones  or  units,  the  second  figure 
tens,  and  the  third  hundreds « 

46.  In  reading  numbers  expressed  by  three  figures,  the 

tens  are  read  after  the  hundreds  without  the  word  and. 

Thus,  478  is  read  four  hundred  seventy-eight,  instead  of  four  hun- 
dred and  seventy-eight.  205  is  read  two  hundred  five,  not  two 
hundred  and  five. 

78 


NOTATION  AND  NUMERATION.  79 

EXERCISES^ 

47.  What  number  is  represented  in  the  picture  on  page  78  ? 
Show  with  objects,  and  tell  what  you  understand  by 

303      121      763      286     431      295     333     107     113     103 
400     200     101      111      321      117     115     240     276     189 

Write  the  following  numbers,  placing  units  under  units, 
tens  under  tens,  etc. .: 
Three  hundred  forty-six.  Sixteen. 

Five  hundred  eighteen.  Six  hundred  eightj^-fiveo 

Ninety-two.  Forty-seven. 

One  hundred  seventy-five.        Seven  hundred  sixteen. 
Two  hundred  eighty.  Two  hundred  two. 

Seven  hundred  eleven.  Four  hundred  eleven. 

Four  hundred  forty-four.  Eight  hundred. 

Three  hundred  five.  Five  hundred  twenty-nine. 

Eight  hundred  seventeen.         Nine  hundred  seventy-three. 
Six  hundred  nine.  Seven  hundred  sixty-four. 

Read  the  numbers  you  have  written. 

How  many  figures  are  needed  to  express  hundreds  ? 

48.  The   method   of   expressing  numbers  by  figures  or 
letters  is  called  Notation. 

•     49.   The  method  of   expressing  numbers  by  figures   is 
called  the  Arabic  Notation. 

It  is  called  the  Arabic  Notation  because  it  was  first  introduced  into 
Europe  by  the  Arabs. 

50.  The  method  of  reading  numbers  is  called  Numeration. 

EXERCISES. 

51.  1.    Write  in  columns  of   ten  numbers  each  all  the 
numbers  from  one  hundred  to  one  hundred  forty-nine. 

2.  From  two  hundred  fifty -one  to  three  hundred. 

3.  What  number  is  next  greater  than  549^  328,  191,  400, 
589,  240,  379,  599,  109,  765  ? 


80  ELEMENTS  OF   ARITHMETIC. 

4.  Make  the  following  numbers  larger  by  one  hundred, 
and  write  them :  17,  89,  30,  4,  52,  305. 

5.  What  number  comes  next  before  100, 700, 201,  870, 310  ? 

6.  How  many  hundreds  are  there  in  each  of  the  folio w= 
ing  numbers  ?     How  many  tens  ?     How  many  units  ? 

468      453      754      902      999      837      964      815 
125      602      882      176      731      395      618      274 

52.   What  is  the  smallest  number  you  can  express  by 
three  figures  ?     The  largest  ? 
999         If  we  add  1  to  999,   we  have  nine  units  and  1, 
1     making  ten  units  or  one  ten.     Adding  the  one  ten  and 
-jr^QQ     the  nine  tens,  w^e  have  ten  tens  or  one  hundred.    Add- 
ing the  one  hundred  to  the  nine  hundreds,  we  have 
ten  hundreds,  which  we  call  by  the  new  name  one  thousand. 
7000  is  read,  seven  thousand.     1720  is  read,  one  thousand  ^even 
hundred  twenty. 

EXERCISES. 

Eead  the  following : 


1.      2. 

3.       4. 

5.      6.       7. 

2000    3000 

5000    7000 

4000    8000    9000 

3791    4809 

7002    9100 

4700    2083    1005 

4263    5074 

2380    1234 

5678    7985    9090 

8.  What  place,  counting  from 

the  right,  do  thousands 

occupy  ? 

9.    What  is  the  smallest  whole  number  that  can  be  ex- 
pressed by  four  figures  ?     What  is  the  greatest  ? 

10.    How  many  units,  tens,  and  hundreds  are  expressed 
by  each  of  the  following  numbers  ? 

Write  and  read  : 

16.  17. 

5055  9100 

7643  1001 

5010  1110 


11. 

12. 

13. 

14. 

15. 

6709 

7891 

7000 

7777 

1098 

7253 

8005 

7700 

4028 

7005 

1389 

6666 

7770 

2003 

4809 

KOTATION   AND   NUMERATION.  81 

18.  Read  the  numbers  in  11,  12,  13,  and  14,  reading  the 
thousands  and  hundreds  together  as  hundreds. 

Thus,  6709,  or  sixty-seven  hundred  nme. 

19.  In  8888,  for  what  does  each  of  the  S's  stand  ? 

20.  What  effect  would  it  have  upon  the  value  of  7  in  71, 
if  a  cipher  were  placed  to  the  right  of  the  1  ? 

53.  When  we  reach  a  thousand,  we  begin  to  count  the 
thousands  as  we  did  the  units,  thus  :  1  thousand,  2  thousand, 
etc.,  up  to  999  thousand,  and  when  we  have  a  thousand 
thousand,  ^  e  call  the  number  one  million. 

A  thousand  million  we  call  a  billion. 
A  thousand  billion  we  call  a  trillion. 

54.  For  convenience,  when  more  than  three  figures  are 
employed  to  express  any  whole  number,  they  are  divided 
into  groups  called  Periods,  the  first  of  which,  counting  from 
the  right,  is  used  to  denote  any  number  from  1  to  999  units; 
the  second,  from  1  to  999  thousands;  the  third,  from  1  to 
999  millions,  etc. 

For  convenience  in  reading,  the  periods  are  usually  sepa- 
rated from  each  other  by  commas. 

Table. 


EL.    OF    AR.  —  0 


82  ELEMENTS   OF  ARITHMETIC. 

The  first  number  in  the  cable  is  read :  seven  hundred  forty-three 
trillion,  nine  hundred  eighty-six  billion,  Jive  hundred  forty-three 
million,  two  hundred  six  thousand,  seven  hundred  ?iineteen. 

Eead  the  foregoing  numbers,  looking  at  tlie  headings 
until  you  become  familiar  with  the  names  of  the  periods. 

55.  Wlieyi  tue  read  numbers,  we  begin  at  the  right  and  sep- 
arate the  numbers  into  periods  of  three  figures  each.  Then, 
beginning  at  the  left,  we  read  each  period  as  if  it  stood  alone, 
adding  its  name. 

Each  period,  except  the  highest  or  left-hand  period,  must  contain 
three  figures. 

EXERCISES. 

66.   Write  and  read  : 

1.                         2.  3.                         4. 

700,000  903,744  300,300  17,019 

965,237  200,080  37,961  586,240 

83,016  147,906  145,808  770,300 

728,000  400,004  307,029  562,999 

5.  6.  7. 

1,789,645  769,834,521  211,467,431       . 

29,783,240  888,971,646  462,293,070 

8.  Erase  the  figure  at  the  right  hand  of  each  number 
in  5,  6,  and  7,  arrange  the  remaining  figures  in  periods 
anew,  by  placing  commas  where  they  belong,  and  then  read 
the  numbers. 

9.  Continue  thus  with  the  given  numbers,  erasing  the 
figures  one  by  one,  pointing  off  into  periods  correctly,  and 
reading  the  numbers. 

10.  Make  the  following  numbers  larger  by  one  thousand. 
By  three  thousand.    By  seven  thousand.    By  nine  thousand. 

763  265  110  703 


NOTATION  AND  NUMERATION.  83 

11.  What  number  is  next  greater  than  1000?  1699? 
2019?  5082?  7200?  1239?  7379?  6299?  9999? 
7676  ?      8109  ?      4099  ? 

12.  What  number  comes  next  before  3000  ?  6000?  5400? 
2040?     2010?     5790?     1000?     2010?     9801? 

13.  How  many  hundreds  and  tens  are  there  in  34  tens? 
In  86  tens  ?  In  37  tens  ?  In  42  tens  ?  In  128  tens  ?  In 
564  tens  ? 

14.  How  many  hundreds  are  there  in  386  units  ?  In  864 
units  ?  In  130  units  ?  In  175  units  ?  In  679  units  ?  In 
534  tens  ?     In  643  tens  ?     In  161  tens  ?     In  879  tens  ? 

15.  How  many  thousands  and  hundreds  are  there  in  75 
hundreds  ?  In  51  hundreds  ?  In  42  hundreds  ?  In  78  hun- 
dreds ?  How  many  in  250  tens  ?  In  175  tens  ?  In  549 
tens  ?     In  543  tens  ?     How  many  in  9875  units  ? 

16.  Make  the  following  numbers  larger,  first  by  ten  thou- 
sand ;  then  by  fifty  thousand ;  then  by  seventy  thousand : 
834        227        386        700         652         200        507        993 

17.  What  number  is  next  greater  than  15,999  ?  150,109? 
199,999?     777,779?     976,190?     275,300?     789,999? 

18.  What  number  is  next  less  than  400,001  ?  600,000  ? 
289,223?  500,000?  943,000?  727,800?  482,176? 
100,000?      400,680?      400,999?      700,010?      300,000? 

WRITTEN   EXERCISES. 
57.   Write  in  figures  : 

1.  Six  thousand,  eight  hundred  fort}^ 

2.  Eight  thousand,  two  hundred  fifty-seven. 

3.  Nineteen  thousand,  five  hundred  twenty-seven. 

4.  Eighty-four  thousand,  six  hundred  thirty-one. 
6.    One  hundred  thousand,  seven  hundred  ten. 

6.  Fifty-eight  thousand,  six  hundred  one. 

7.  Two  hundred  thousand,  four  hundred  seventy-five. 

8.  Four  hundred  eighteen  thousand,  two  hundred  seven. 


84  ELEMENTS  OF   ARITHMETIC. 

9.    Five  hundred  seven  thousand,  thirty. 

10.  Seven  hundred  fifteen  thousand,  nine  hundred  twenty-, 
four. 

11.  Thirty-two  million,  four  hundred  fifteen  thousand^ 
two  hundred  thirty -three. 

12.  One  hundred  six  million,  eight  hundred  nine  thou- 
sand, seven  hundred  fifty. 

13.  Two  hundred  sixty-five  million,  fifteen  thousand,  one 
hundred  seventy-nine. 

14.  Three  hundred  sixty -five  million,  forty-nine. 

15.  Nine  million,  nine  thousand,  nine. 

16.  Eighty  million,  eighty  thousand,  eighty. 

17.  Six  hundred  million,  six  hundred  thousand,  six  hun> 
dred. 

18.  Thirty-six  million,  eighteen  thousand. 

19.  Twenty-four  million,  eight  thousand,  eight. 

20.  Two  hundred  million,  two  thousand,  two. 

21.  Thirty-nine  billion,  sixty   million,    sixty   thousand, 
sixty. 

22.  Fifty-two  billion,  one  million,  one  thousand,  one. 

23.  Sixty  billion,  sixty  million,  sixty  thousand,  sixty. 

24.  Eight  hundred  million,  eighty-four  thousand. 

25.  Seventy-seven  billion,  seventy-seven  million,  seventyN 
seven  thousand,  seventy-seven. 

26.  Four  hundred  twenty-six  million,  four  hundred  thou- 
sand, twenty-five. 

27.  One  hundred  seventy  million,  seven. 

28.  Sixteen  hundred  seventy-eight. 

29.  One  trillion,  seventeen  million. 

30.  Four  hundred  fifty-six  million,  ten. 

31.  Five  hundred  nine  million,  four  hundred  thousand, 
twenty-five. 

32.  Nineteen  hundred  two. 

33.  One  billion,  six  hundred  thousand,  four. 


NOTATION   AND   NUMERATION.  85 

NOTATION   AND   NUMERATION  OF  UNITED 
STATES  MONEY. 

58.  Write  the  names  of  all  the  United  States  coins  you 
have  seen,  and  tell  of  what  each  is  made.  What  paper 
money  have  you  seen? 

In  the  currency  of  the  United  States, 

10  mills  make  1  cent. 
100  cents  make  1  dollar. 

59.  The  Sign  of  Dollars  is  ^.  It  is  written  before  the 
number. 

Thus,  $27  is  read,  twenty-seven  dollars. 

60.  In  writing  cejits  and  mills,  a  period  called  the  Decimal 
Point  is  placed  before  the  number  of  cents. 

61.  Cents  occupy  the  Jirst  tico  places  at  the  right  of  the 
decimal  point,  and  7nills  the  third  place. 

Thus,  .$0.35  is  read,  thirty-five  cents;  $2,875  is  read,  two  dollars 
eighty-seven  cents  five  mills. 

Mills  are  not  coined.  In  business  five  mills  or  more  are  called  one 
cent,  and  less  than  five  are  disregarded. 

62.  If  the  number  of  cents  is  less  than  ten,  a  cipher  must 
be  written  in  the  first  place  at  the  right  of  the  decimal  point. 

Thus,  five  cents  is  written  $.05;  three  cents,  $.03;  seven  dollars 
two  cents  five  mills  is  written  $  7.025. 

EXERCISES. 

63.  Read  the  following : 

1.  2.                    3.                     4.                   5. 

^1.00  ^1.75  ^1.65  33.185  $0.25. 

$4.87  $18.05  $15.43  $70.10  $1.05. 

$3.04  $24.16  $100.50  $9,125  $   .40. 

$7.  $2,357  $10.  $0,375  $7.20. 


86  ELEMENTS   OF  ARITHMETIC. 

Write  the  following  in  columns^  keeping  decimal  points 
in  a  vertical  line  : 

1.  Eighteen  dollars  twenty-four  cents. 

2.  Twenty-six  dollars  thirty-six  cents. 

3.  Fifty-one  dollars  five  cents  five  mills. 

4.  Thirty-eight  dollars  sixteen  cents  three  mills. 

5.  Thirty-nine  dollars  nine  cents. 

6.  Thirty  dollars  five  cents. 

7.  One  hundred  dollars  ten  cents  four  mills. 

8.  Six  dollars  six  cents. 

9.  Write  one  cent,  two  cents,  three  cents,  etc.,  to  fifty 
cents. 

10.  How  many  cents  are  there  in  f  1?     In  ^  2  ?     In  $3  ? 
Inf4?     In  $8?     In  $5?     In  $9?     In  $7?    In  ^6? 

11.  Eead  as  cents:    $2;    ^2.02;    $3;    $3.05;    $3.75; 
$4.17;  $9.08;  $11.11. 

12.  Write  the  following  as  cents,  using  the  character  ^ 
for  cents : 

$1.25        $1.10        $3.75        $7.45        $9.67        $8.80 
$7.05        $9.81        $4.13        $4.18        ^5M        $6.72 

13.  Eead  the  following  as  dollars  and  cents  : 
Thus,  1728j^  =  17  hundred  28  cents,  or  17  dollars  28  cents. 

2375^        300^        1015^  1005^  680p        2793^ 

1390^        970^  700)^        57345^        3250^        1809;^ 

14.  Write  the  same,  using  the  sign  of  dollars. 

15.  How  many  mills  are  there  in  2^?     In  5^?     In  7^? 
In  9^  ?     In  10^  ?     In  70^  ?     In  30^  ?     In  90^  ?     In  15^  ? 

16.  How  many  mills  are  there  in  lOOj^  ?     In  300^?     In 
$1?     In  $2?     In  $9?     In  $1.20?     In  $1.50? 

17.  Eead  the  following  as  dollars,  cents,  and  mills : 
2765  mills  5639  mills  2793  mills 

235  mills  9050  mills  4505  mills 

18»    Write  the  above,  using  the  dollar  sign. 


NOTATION   AND   NUMERATION.  87 


ROMAN  NOTATION. 

64.  The   method   of   expressing   numbers  by  means  of 
letters  is  called  Roman  Notation. 

It  is  called  Roman  Notation  because  it  was  originally  used  by  the 
ancient  Romans. 

65.  In  this  system,   seven  capital  letters   are   used  to 
express  numbers,  viz. : 

Letters:         I,      V,      X,         L,  C,  D,  M. 

Values:  1,       5,      10,      50,      100,      500,      1000. 

By  combining  these  letters  according  to  certain  principles, 
any  number  can  be  expressed. 

66.  Principles.  —  1.    Wien  a  letter  is  repeated,  its  value 
is  repeated. 

Thus,  X  represents  ten ;  XX,  twenty  ;  XXX,  thirty  ;  CC,  two  hun- 
dred ;  MM,  two  thousand. 

2.  When  a  letter  is  jjlaced  before  another  of  greater  value, 

its  value  is  to  be  taken  from  that  of  the  greater. 

Thus,  I  represents  one,  V,  five,  and  X,  ten.    But  IV  represents 
four  ;  IX,  nine  ;  XL,  forty  ;  XC,  ninety. 

3.  When  a  letter  is  placed  aftei'  another  of  greater  value, 

their  values  are  to  be  united. 

Thus,  XV  represents  fifteen  ;  LXX,  seventy  ;  LXXX,  eighty  ;  MC, 
one  thousand,  one  hundred. 

4.  When  a  letter  is  placed,  between  tivo  letters,  each  of  greater 

value  than  itself,  its  vcdue  is  to  be  taken  from  the  smn  of  the 

other  two. 

Thus,  XIV  represents  fourteen ;   XIX,  nineteen ;   CIV,  one  hun- 
dred four. 

5.  A  bar  placed  over  a  letter  increases  its  vcdue  a  thousand 
fold. 

Thus,  V  represents  five  ;  V.  five  thousand  ;  LXX  represents  seventy  ; 
LXX,  seventy  thousand. 


88 


ELEMENTS   OF   ARITHMETIC. 


Table. 


1=  1 

XIII  =  13 

LX: 

=     60 

11=    2 

XIV  =  14 

LXX: 

=     70 

111=    3 

XV  =  15 

XC: 

=     90 

TV=    4 

XVI  =  16 

C: 

=    100 

V=    5 

XIX  =  19 

cc 

=    200 

VI  =    6 

XX  =  20 

cccc 

=    400 

VII  =    7 

XXI  =  21 

CD 

=    400 

VIII  =    8 

XXIX  =  29 

D 

=    500 

IX  =    9 

XXX  =  30 

DCC 

=    700 

X  =  10 

XXXIV  =  34 

M 

=  1000 

XI  =  11 

XL  =  40 

MMM 

=  3000 

XII  =  12 

L  =  50 
EXERCISES. 

MDCCCXCII 

=  1892 

Eead  the  following : 

XX 

XIX            CDXX 

XLIV              XXV 

XXXV             LXIV 

XL 
XC 

XCIV 

LXX 

XIX 

XCIX 

VIII 

MDLIV 

XXI 

LXIX       CCXXIV 

LX 

MDCCC 

XLIX 

XXXVI        CCCLIX 

DCXL 

MMDC 

MDCCCX 

CCXXVI         DLXXI 

MDXL 

DCCL 

cxcv 

CCXLIV         IMMMD 

IVDXL 

CCXCV 

Express  the  following  by  Eoman  Notation  : 

23       61 

84      35      312       51 

7       1010 

1900 

34  19  59  47  419  493  1800  8000 
15  36  62  86  226  499  1492  9000 
27   43   97  214   384   278   1607   1721 


of 


ADDITION. 


67.  1.  How  many  blocks  are  5  blocks,  3  blocks,  and  7 
blocks  ? 

2.  How   many   splints    are    3   splints,  2   splints,  and  8 
splints  ? 

3.  How  many  oranges  are  7  oranges,  2  oranges,  and  5 
oranges  ? 

4.  How  many  are  5  and  6  and  7  ? 
6.    How  many  are  7  and  8  and  6  ? 

6.  How  many  are  5  and  9  and  2  and  6  ? 

7.  How  many  are  3  and  7  and  4  and  8  ? 

8.  How  many  are  4  and  8  and  5  and  9  ? 

9.  What  have  you  been  doing  with  the  numbers  given 
above  ? 

68.  The  process  of  finding  a  number  that  is  equal  to  two 
or  more  given  numbers  is  called  Addition. 

69.  The  result  obtained  by  adding  is  the  Sum,  or  Amount. 

70.  The  Sign  of  Addition  is  a  small  upright  cross  (+). 

It  is  read  j)lus,  and  is  placed  between  the  numbers  to  be 

added. 

Thus,  4  +  2  is  read  4  plus  2,  and  means  that  4  and  2  are  to  be 
added. 

71.  The  Sign  of  Equality  is  two  short  parallel  horizontal 
lines  (  =  ).     It  is  read  equals,  or  is  equal  to. 

Thus,  3  +  2  =  5  is  read  3  plus  2  equals  5. 


90  ELEMENTS  OF  ARITHMETIC. 

WRITTEN   EXERCISES. 
72.   1 .    Add  13,  12,  24,  30,  and  14. 


Explanation.  —  In  adding,  the  numbers  are  written  in 
columns,  units  under  units,  tens  under  tens,  and  it  is  more 
12  convenient  to  begin  with  the  units'  column.  Adding,  we  find 
24  there  are  13  units.  13  means  1  ten  and  3  units  ;  therefore  the  3  is 
OA  written  in  the  units'  column,  and  the  1  ten  is  added  to  the  tens. 
Adding  the  tens,  we  find  that  there  are  9  tens.  Therefore  the 
^     result  is  93. 

QQ  In  adding,  do  not  say  4  and  4  are  eight  and  2  are   ten. 

etc. ,  simply  name  results,  thus :  4,  8,  10,  13. 


13 


Find  the  sum 

of 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

72 

73 

84 

57 

^27 

$21 

$96 

$20 

85 

97 

17 

46 

72 

53 

81 

38 

90 

16 

52 

93 

18 

38 

45 

47 

36 

68 

78 

16 

19 

73 

78 

93 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

17. 

39 

54 

89 

2S 

59 

2^ 

45 

68 

29 

90 

24 

11 

83 

27 

17 

39 

72 

71 

74 

50 

17 

48 

18 

74 

45 

29 

93 

33 

26 

16 

13 

m 

13 

36 

35 

^Q 

28 

36 

14 

23 

18. 

19. 

20. 

21. 

22. 

23. 

24. 

25. 

45 

62 

42 

Q5 

43 

Q>^ 

bQ 

42 

58 

87 

24 

89 

69 

42 

49 

46 

34 

49 

85 

38 

22 

25 

17 

34 

22 

36 

48 

76 

^^ 

34 

21 

5S 

67 

55 

34 

45 

74 

28 

36 

91 

ADDITION.  91 

26.  Two  birds  fly  in  opposite  directions,  one  19  miles, 
and  the  other  27  miles.     How  far  apart  are  they  ? 

27.  Max  earned  24  dollars,  and  John  earned  9  dollars 
more  than  Max.     How  many  dollars  did  both  earn  ? 

28.  Baby  is  4  years  old ;  mamma  is  seven  times  as  old 
as  baby  ;  grandma  is  twice  as  old  as  mamma.  What  is  the 
sum  of  their  ages  ? 

29.  In  a  store  window  I  saw  38  books,  25  toys,  17  slates, 
and  20  globes.     How  many  articles  did  I  see  ? 

30.  In  one  class  in  our  school  there  are  11  boys  and  9 
girls ;  in  another,  17  boys  and  18  girls  ;  in  another,  10  boys 
and  14  girls.  How  many  boys  are  there  in  all  ?  How 
many  girls  ?     How  many  children  ? 

31.  A  fisherman  caught  46  trout,  75  bass,  and  13  blue- 
fish.     How  many  fishes  did  he  catch  ? 

32.  Fred  weighs  83  pounds,  and  Nathan  weighs  97  pounds. 
How  much  do  they  both  weigh  ? 

33.  In  one  book  there  are  98  pages,  in  another  89,  and 
in  a  third  75.     How  many  pages  are  there  in  all  the  books  ? 

34.  A  certain  tract  of  land  was  divided  into  4  farms; 
one  containing  13  acres,  another  37  acres,  the  third  70  acres, 
and  the  fourth  48  acres.  How  many  acres  did  the  original 
tract  contain  ? 

35.  The  distance  from  Alburn  to  Grade  is  6  miles,  from 
Grade  to  Newton  8  miles,  from  Newton  to  Arnold  9  miles, 
and  from  Arnold  to  Houston  8  miles.  How  far  is  it  from 
Alburn  to  Houston  ? 

36.  Elsie  solved  18  examples  on  Monday,  14  on  Tues- 
day, and  12  on  Wednesday.  How  many  examples  did  she 
solve  in  all  ? 

37.  If  I  buy  a  collar  for  25  cents,  some  gloves  for  37 
cents,  some  ribbon  for  8  cents,  and  a  paper  of  needles  for  9 
cents,  what  do  they  all  cost  ? 

38.  A  farmer  has  in  one  bin  28  bushels  of  oats,  in  another 


92  ELEMENTS   OF   ARITHMETIC. 

36,  in  another  23.     How  many  bushels  has  he  in  the  three 
bins  ? 

39.  If  I  sell  8  chickens  to  one  man,  18  to  another,  and 
have  24  left,  how  many  had  I  at  first  ? 

40.  Mr.  Atwater  bought  a  cow  for  36  dollars,  a  sheep 
for  8  dollars,  and  a  calf  for  13  dollars.  How  much  did  he 
pay  for  all  ? 

41.  Susie  has  38  cents,  and  her  brother  has  17  cents  more 
than  she  has.     How  many  cents  have  they  both  ? 

42.  Morton's  geography  cost  75  cents,  his  reader  39 
cents,  and  his  slate  18  cents.    How  much  did  they  all  cost  ? 

43.  Mr.  Barnes  has  3  fields.  In  one  he  put  36  cows,  in 
another  42  cows,  and  in  the  third  27  cows.  How  many 
cows  has  he  in  the  three  fields  ? 

44.  A  fisherman  sold  4  fishes  for  the  following  sums  :  the 
first  for  37  cents,  the  second  for  44  cents,  the  third  for  54 
cents,  the  fourth  for  48  cents.  What  did  he  receive  for 
all? 

45.  On  a  Christmas  tree  I  counted  12  dolls,  25  balls,  28 
flags,  5  whistles,  3  drums,  25  candy -bags,  and  14  -books. 
How  many  things  did  I  count  ? 

46.  Thornton  has  68  marbles,  Oliver  has  10  less  than 
Thornton.     How  many  have  both  boys  together  ? 

47.  One  stalk  of  a  plant  has  43  leaves,  a  second  stalk 
has  36  leaves,  and  a  third  has  18  leaves.  How  many  leaves 
have  they  all  ? 

48.  A  baker  made  at  one  time  37  loaves,  at  another  56, 
at  another  28.     How  many  loaves  did  he  make  in  all  ? 

49.  A  milkman  sold  72  quarts  of  milk  on  Monday,  56  on 
Tuesday,  85  on  Wednesday,  and  49  on  Thursday.  How 
many  quarts  did  he  sell  in  the  four  days  ? 

50.  A  butcher  sold  24  pounds  of  veal,  37  pounds  of  porkj 
38  pounds  of  chickens,  and  49  pounds  of  beef.  How  many 
pounds  of  meat  did  he  sell  ? 


ADDITION,  93 

51.  A  mail  has  45  sheep,  28  cows,  and  15  calves.  How 
many  animals  has  he  ? 

62.  If  I  spend  8  dollars  a  week  for  board,  3  dollars  for 
car  fare,  and  have  14  dollars  left  of  my  week's  earnings, 
how  much  did  I  earn  ? 

53.  Wilfred  paid  28  dollars  for  a  desk.  He  paid  5  dol- 
lars to  have  it  polished,  and  then  sold  it  for  7  dollars 
more  than  it  cost  him.  How  much  dil  he  get  for  the 
desk? 

54.  Commencing  with  28,  find  the  sum  of  28  and  the 
succeeding  numbers  less  than  36  ? 

55.  A  cloth  merchant  sold  5  rolls  of  cloth,  containing 
respectively  46,  52,  63,  29,  and  76  yards.  How  many  yards 
did  he  sell  ? 

56.  A  saddle  cost  14  dollars,  a  wagon  60  dollars  more 
than  the  saddle,  and  a  horse  as  much  as  the  saddle  and 
wagon  together  How  much  did  the  horse  cost?  How 
much  did  all  cost  ? 

57.  Harold  earned  87  cents.  If  he  previously  had  36 
cents,  and  then  found  7  cents,  how  many  cents  had  he  in 
all? 

58.  Howard  wrote  38  lines  of  German  on  Monday,  27  on 
Tuesday,  15  on  Wednesday,  17  on  Thursday,  and  26  on 
Friday.     How  many  lines  did  he  write  during  the  week  ? 

59.  A  man  sold  three  lots.  For  one  he  received  $  695, 
for  another  $  734,  and  for  the  third  $  392.  How  much  did 
he  receive  for  all  ? 

60.  A  owns  a  farm  of  518  acres,  B  one  of  425  acres,  and 
C  one  of  392  acres.     How  many  acres  do  they  all  own  ? 

61.  In  a  city  there  were  four  schools,  the  first  containing 
396  pupils,  the  second  683  pupils,  the  third  718  pupils, 
and  the  fourth  as  many  as  the  second  and  third  together. 
How  many  pupils  attended  the  schools  ?  How  many  at- 
tended the  fourth  school  ? 


94  ELEMENTS  OF  ARITHMETIC. 


Add  the  following : 

62. 

63. 

64. 

65. 

66. 

67. 

68. 

68 

54 

37 

81 

39 

55 

38 

45 

32 

93 

39 

23 

38 

43 

93 

81 

86 

44 

87 

37 

71 

87 

36 

94 

87 

65 

25 

64 

94 

57 

83 

39 

34 

61 

83 

86 

91 

27 

86 

56 

39 

73 

83 

87 

55 

54 

83 

24 

98 

37 

32 

66 

37 

74 

86 

72 

69. 

70. 

71. 

72. 

73. 

74. 

452 

374 

418 

519 

735 

875 

378 

516 

372 

327 

872 

912 

684 

834 

384 

684 

659 

386 

359 

912 

596 

519 

738 

596 

871 

816 

387 

329 

486 

837 

964  ., 

325 

426 

548 

684 

542 

832 

426 

385 

819 

396 

816 

576 

845 

713 

384 

845 

347 

75. 

76. 

77. 

78. 

79. 

3785 

7351 

7638 

6835 

381 

4651 

4286 

1842 

7867 

3468 

3278 

3875 

3451 

392 

193 

5964 

1894 

4265 

8437 

8200 

8137 

5768 

7169 

24 

57 

8462 

3681 

8376 

6174 

93 

5738 

5437 

1452 

5762 

3956 

9124 

8796 

3814 

8600 

4184 

6783 

5106 

6842 

439 

324 

3854 

7340 

6937 

3764 

47 

4265 

4165 

5843 

158 

596 

7384 

9354 

7165 

99 

1938 

SUBTRACTION. 


73.  1.  How  many  cents  has  George  if  he  lost  5  cents  of 
the  9  cents  which  he  had  ? 

2.  Ten  horses  were  drawing  wagons,  all  but  5  of  which 
were  gray.     How  many  gray  horses  were  there  ? 

3.  Bertha  had  13  cents  and  spent  6  cents.  How  much 
had  she  left  ? 

4.  How  many  are  10  less  3  ?     15  less  8  ?     17  less  7  ? 

5.  How  many  are  16  less  7  ?     19  less  9  ?     13  less  5  ? 

6.  How  many  are  12  less  8  ?     14  less  5  ?     15  less  7  ? 

7.  How  many  are  13  less  6  ?     17  less  9  ?     18  less  9  ? 

8.  What  have  you  been  doing  with  the  above  numbers  ? 

74.  The  process  of  finding  the  difference  between  two 
numbers,  or  of  taking  part  of  a  number  from  it  and  finding 
how  many  are  left,  is  called  Subtraction. 

75.  The  number  from  which  another  is  taken  is  called 
the  Minuend. 

76.  The  number  to  be  subtracted  is  the  Subtrahend. 

77.  The  number  remaining  after  one  number  is  taken 
from  another  is  called  the  Remainder. 

The  remainder  is  the  difference  between  the  minuend  and 
the  subtrahend. 

78.  The  Sign  of  Subtraction  is  a  short  horizontal  line 
(  — ).     It  is  called  Minus. 

Thus,  8  —  3  is  read,  8  minus  3,  and  means  that  3  is  to  be  subtracted 
from  8. 

95 


ELEMENTS   OF   ARITHMETICo 


WRITTEN    EXERCISES. 


79.    1.    From  869  subtract  423. 


Explanation.  —  For  convenience,  the  less 
number  is  written  under  the  greater,  units  under 
units,  tens  under  tens,  etc.  Beginning  at  the  right, 
each  order  of  units  in  the  subtrahend  is  subtracted 
separately  from  the  same  order  in  the  minuend. 
3  units  are  6  units,  which  are  written  under  units. 


Minuend        869 
Subtrahend  423 

Remainder    446 

Thus,  9  units  - 

6  tens  —  2  tens  are  4  tens,  which  are  written  under  the  tens. 

8  hundreds  —  4  hundreds  are  4  hundreds,  which  are  written  under 
the  hundreds.     Hence  the  remainder  is  446. 

Proof. — 446,  the  remainder,  plus  423,  the  subtrahend,  equals  869, 
the  minuend.     Hence  the  result  is  correct. 


Copy,  subtract,  and  prove  : 


2. 

3. 

4.      5. 

6. 

7. 

8. 

888 

999 

564     899 

329 

957 

864 

356 

276 

213     486 

115 

516 

231 

9. 

10. 

11.      12. 

13. 

14. 

15. 

435 

768 

975     543 

879 

768 

599 

234 

253 

432     133 

217 

642 

*  143 

16. 

17. 

18. 

19. 

20. 

From 

876 

789 

968 

999 

888 

Take 

242 

176 

723 

275 

234 

21.  Susie  had  two  bundles  of  10  splints  each,  and  6  more. 
How  many  splints  had  she  ?  She  gave  her  teacher  8  of  the 
splints.     How  many  had  she  left  ? 

She  found  the  6  ones  were  not  enough  to  give 
her  teacher,  so  she  took  one  bundle  of  the  tens, 
untied  it,  and  put  it  with  her  6  ones,  making  16 
ones  besides  the  1  ten  which  she  had  not  untied. 
She  took  8  away  from  the  16  and  found  she  had  8 
left,  and  also  the  1  ten,  or  18  left  in  all. 

Do  just  as  Susie  did. 


116 

%^ 
8 

18,  Remainder. 


SUBTRACTION.  97 

22.  Take  1  bundle  of  one  hundred  splintS;  4  bundles  of 
10  splints  each,  and  5  single  splints.  How  many  have  you 
in  all?  Give  your  teacher  27  of  the  splints.  Compare 
what  you  did  with  the  following  solution: 

3  15 

14p 

2  7 


118,  Remainder. 

23.    Annie  had  2  bundles  of  one  hundred  splints  eachc 
She  gave  Fred  78.     How  many  had  she  left  ? 

Annie  took  1  bundle  of  one  hundred,  untied  it,  and  found 
there  were  10  tens  in  it.  She  left  9  of  the  tens  a  moment  and 
untied  1  bundle  of  10,  finding  10  ones  in  it.  Of  these  she 
gave  8  to  Fred,  and  had  2  left.  Then  taking  up  the  9  tens, 
she  gave  7  tens  to  Fred,  and  had  2  tens  left,  and  besides 
these  the  1  bundle  of  one  hundred.  She  had,  therefore, 
122  left. 

Eind  the  difference  between  the  following,  and  supply  the 
numbers  omitted  in  minuend,  subtrahend,  or  remainder : 


24. 

25. 

26. 

27. 

28. 

29. 

30. 

31. 

34 

86 

92 

41 

70 

83 

44 

96 

17 

69 

55 

18 

29 

45^ 

28_ 

73 

32. 

33. 

34. 

35. 

36. 

37. 

38. 

39. 

99 

72 

77 

77 

91 

85 

88 

65 

45 

48 

43 

29 

88 

17 

33 

28. 

40. 

41. 

42. 

43. 

44. 

45. 

46. 

47. 

871 

761 

240 

166 

752 

631 

382 

546 

257 

347 

18 

139 

239 

408 

272 

338 

48. 

49. 

50. 

51. 

52. 

53„ 

54. 

Minuend 

^ 

300 

1   ] 

204 

500 

607 

908 

550 

600 

Subtrahend, 

171 

78 

183 

179 

127 

258 

379 

Remainder, 

EL. 

or 

AK.- 

-7 

98 


ELEMENTS   OF   ARITHMETICo 


65.          56. 

57.         58. 

59.          60. 

61. 

Minuend, 

635       591 

624       350 

870       980 

527 

Subtrahend, 

469       268 

156       123 

456       379 

318 

Remainder, 

62.            63. 

64.            65, 

66. 

67. 

Minuend, 

?            769 

?            780          800 

? 

Subtrahend, 

17            ? 

238            ? 

9 

502 

Remainder, 

136          144 

196          123          191 

139 

68. 

69. 

70. 

Minuend, 

$541 

$905 

768  children. 

Subtrahend, 

$•492 

$184 

599 

Remainder, 

$ 

$ 

children. 

71. 

72. 

73. 

Minuend, 

808  eggs. 

967  birds. 

842  pints. 

Subtrahend, 

527    " 

209     " 

763     " 

Remainder, 

eggs. 

birds. 

pints. 

74o 

75. 

76. 

Minuend, 

400  horses. 

502  cows. 

$550 

Subtrahend, 

285       " 

237     " 

$118 

Remainder, 

horses. 

cows. 

$ 

77. 

78. 

79. 

Minuend, 

800  bushels. 

?    apples. 

?   days» 

Subtrahend, 

269       " 

372       " 

18     " 

Remainder, 

bushels. 

263  apples. 

347  daya 

80. 

81. 

82. 

Minuend, 

75  years. 

$772 

$324 

Subtrahend, 

?       " 

$624 

9 

Remainder, 

19  years. 

$ 

$136 

SUBTRACTION  99 

REVIEW  EXERCISES. 
ADDITION    AND    SUBTRACTION. 
80.    1.    From  a  tank  containing  935  gallons  of  water,  648 
gallons  were  drawn  off,  after  wMch.  247  gallons  ran   in. 
How  many  gallons  were  there  then  in  the  tank  ? 

2.  Two  trains  start  from  Chicago.  One  goes  east  298 
miles,  and  the  other  west  247  miles.  How  far  apart  are 
they  ? 

3.  A  man  purchased  8983  bricks,  but  used  only  5362. 
How  many  had  he  left  ? 

4.  A  man  paid  ^465  for  a  carriage.  He  spent  $  57  for 
repairs  on  it,  and  sold  it  for  $  500.  Did  he  gain  or  lose, 
and  how  much  ? 

5.  From  the  sum  of  763  and  297  subtract  their  differ- 
ence. 

6o  A  horse  and  cow  together  cost  $276.  If  the  cow 
cost  f  81,  what  was  the  cost  of  the  horse  ? 

7.  Two  trains  start  from  cities  582  miles  apart.  If 
they  travel  towards  each  other,  one  123  miles  and  the  other 
238  miles,  how  far  apart  are  they  ? 

8.  Columbus  discovered  America  in  1492,  How  many 
years  have  passed  since  then  ? 

9.  A  merchant  bought  30  pieces  of  cloth,  containing  in 
all  979  yards.  He  afterwards  sold  all  but  143  yards.  How 
many  yards  did  he  sell  ? 

10.  The  cost  of  building  a  house  was  $3972,  and  of  fur= 
nishing  it  $  1597.  What  was  the  entire  cost  ?  How  much 
more  did  it  cost  to  build  it  than  to  furnish  it  ? 

11.  Cotton  was  first  planted  in  the  United  States  about 
1759.     How  long  ago  was  that  ? 

12.  Ealph  threw  a  ball  37  feet,  and  Jamie  threw  his  19 
feet  farther  than  Ralph.  How  many  feet  must  Jamie  go  to 
pick  up  his  ball  and  return  ? 


100  ELEMENTS   OF   ARITHMETIC. 

13.  A  man  OAvnecl  a  horse  that  cost  $852,  and  a  carriage 
that  cost  $437.  He  was  obliged  to  sell  them  both  for 
$  1000.     How  much  money  did  he  lose  by  the  sale  ? 

14.  My  neighbor  has  in  his  orchard  264  pear  trees,  which 
is  187  less  than  the  number  of  his  apple  trees.  How  many 
apple  trees  has  he  ?     How  many  trees  has  he  in  all  ? 

15.  A  farmer  having  425  bushels  of  wheat,  sold  127 
bushels  to  one  person,  and  264  bushels  to  another.  How 
many  bushels  had  he  left  ? 

16.  $24.03  +  $17.75  +  $19.63  +  $10  +  $18.09  +  |7.99=? 

17.  $10 -($2.35 +  $3.78 +  $!)  =  ? 

18.  Mr.  Jones  earned  during  July  $  172.  He  paid  a  bill 
for  $  28  to  the  grocer,  $  19  to  the  butcher,  and  $  36  for  rent. 
How  much  of  the  money  remained  ? 

19.  $4700- ($462 +  $2180)=? 

$  3900  -  ($  2614  +  $  854.50)  =  ? 

20.  A  man  gave  to  his  son  $  575 ;  to  his  daughter,  $468; 
and  to  his  nephew,  $249  less  than  to  his  daughter.  How 
much  did  he  give  to  all  ? 

21.  Two  men,  who  are  425  miles  apart,  travel  towards 
each  other,  one  36  miles,  the  other  58  miles  a  day.  How 
far  apart  will  they  be  at  the  end  of  two  days  ? 

22.  If  the  same  men  travel  away  from  each  other,  how 
far  apart  will  they  be  at  the  end  of  two  days  ? 

23.  In  a  certain  school  there  are  692  pupils.  In  the 
primary  grade  there  are  325;  in  the  intermediate  grade, 
259:  and  the  remainder  are  in  the  grammar  grade.  How 
many  are  in  the  grammar  grade  ? 

24.  A  man  bought  a  lot  of  land  for  $  1860,  and  built  upon 
it  a  house  costing  $7584.  If  he  sold  the  property  for 
$  10,000,  how  much  did  he  make  ? 

25.  If  I  buy  at  a  grocery  store  sugar  for  78  cents,  cheese 
for  39  cents,  and  nuts  for  25  cents,  giving  in  payment  a  $2 
bill,  how  much  change  should  I  receive  ? 


SUBTRACTION.  101 

26.  From  946  subtract  486 ;  from  the  result  subtract  284 ; 
to  this  add  398;  and  from  this  answer  subtract  199. 

^27.  Lena's  book  has  482  pages.  If  she  reads  168  pages 
in  the  morning,  and  89  pages  in  the  afternoon,  how  many 
more  pages  has  she  to  read  ? 

28.  Bert  and  Harry  had  each  $1.40.  Bert  earned  28j^ 
more,  and  Harry  spent  12^  of  his  money.  How  much  more 
has  Bert  than  Harry  ? 

29.  In  a  school  there  were  278  girls  and  215  boys.  If  29 
girls  and  34  boys  leave,  how  many  pupils  will  there  then  be 
attending  the  school  ? 

30.  Into  a  bin  that  holds  825  bushels,  I  put  272  bushels 
of  corn,  and  afterwards  346  bushels  more.  How  many  more 
bushels  of  corn  will  it  hold  ? 

31.  If  I  buy  a  hat  for  $  6.75  and  give  in  payment  a  $  10 
bill,  what  change  should  I  receive  ? 

32.  The  entire  expenses  of  a  journey  made  by  a  family 
were  $391.86.  The  fare  on  the  railroads  was  $192.15, 
carriage  hire  was  $  37.25,  fare  on  the  steamers  amounted  to 
$61.35,  and  the  rest  was  paid  for  board.  What  did  the 
board  cost  ? 

33.  A  merchant  failed  in  business,  owing  debts  to  the 
amount  of  $  1837.85.  His  goods  on  hand  were  worth 
$894.50,  and  there  was  due  him  from  persons  who  had 
bought  goods  from  him,  $  591.75.  Hoav  much  more  did  he 
owe  than  he  could  pay  ? 

34.  $  68.95  +  $  81.36  -  $  29.15  -  $  42.36  +  $  25.84  =  ? 

35.  $  76.37  +  $  34.93  +  $  8.83  -  $  16.25  -  $  13.24  =  ? 

36.  $  59.23  +  $  18.76  -  $  32.17  +  $  7.08  -  $  41.25  =  ? 

37.  $  32.94  -  $  16.23  +  $  13.81  -  $  20.06  +  $  24.53  =  ? 

38.  $  68.92  +  $  23.75  -  $  15.27  -  $  6.35  +  $  15.18  =  ? 

39.  $  37.50  -  $  18.24  +  $  60.08  -  $  18.32  -  $  16.29  =  ? 

40.  $51.23 +  $67.      +  $59.38 -$62.81-$  4.85  =  ? 

41.  $  24.69  +  $  41.80  -  $  23  +  $  27.42  _$8  +  $15  =  ? 


MULTIPLICATION. 


81.  1.  Henry  worked  5  days  per  week  for  5  weeks. 
How  many  days  did  he  work  in  that  time  ? 

2o    A  boy  delivered  books  to  12  houses^  leaving  6  books 
at  each  house.     How  many  books  did  he  carry  altogether  ? 

3.  The  wages  of  a  laborer  were  $12  per  week.     How 
much  could  he  earn  in  6  weeks  ? 

4.  The  price  of  a   cord  of  wood   is   $5,     How  much 
must  be  paid  for  7  cords  ? 

5.  The  postman  who  delivers  our  mail  travels  9  miles 
per  day.     How  far  will  he  travel  in  8  days  ? 

6»    A  lady  bought  11  yards  of  cambric  at  10  cents  per 
yard.     How  much  did  it  cost  her  ? 

7.  A  good  walker  can  walk  4  miles  per  hour.      How 
far  can  he  walk  in  9  hours  at  that  rate  ? 

8.  How  many  are  eight  3's  ?     Nine  5's? 

9.  How  many  are  seven  7^s  ?     Eight  6's  ? 

10.  How  many  are  six  8's  ?     Nine  4's  ? 

11.  How  many  are  four  6's  ?     Five  4's  ? 

12.  What  have  you  been  doing  with  the  numbers  in  the 
examples  given  above  ? 

82.  A  short  process  of  finding  the  sum  of  several  equal 
numbers,  or  the  process  of  taking  one  numbsr  as  many  times 
as  there  are  units  in  another  is  called  Multiplication. 

83.  The  number  taken  or  multiplied  is  called  the  Multi- 
plicand. 

102 


MULTIPLICATION. 


103 


84.  The  number  which  shows  how  many  times  the  multi- 
plicand is  taken  is  called  the  Multiplier. 

85.  The  result   obtained   by   multiplying  is  called   the 
Product. 

86.  The  multiplicand  and  multiplier  are  called  the  Factors 
of  the  product. 

87.  The  Sign  of  Multiplication  is  an  oblique  cross,  x  .     It 

is  read  multiplied  by  when  the  multiplicand  precedes  it  and 

times  when  the  multiplier  precedes  it. 

Thus  6  X  8  is  read  6  multiplied  by  8  wlien  6  is  the  multiplicand, 
but  it  is  read  6  times  8  wlien  6  is  tlie  multiplier. 


MULTIPLICATIOM 

TABLE. 

1 

2 

3 

4 

5 

6 

7 

8  1  9 

10 

11 

12 

2 
3 

4 

6 

6 
9 

8 
12 
16 

10 
15 

20 
25 

12 
18 
24 
30 

14 
21 

28 
35 

16 
24 

32 
40 

48 

18 



27 

36 

45 

54 

20 
30 
40 
50 
60 

22 
33 
44 

55 

m 

24 
36 
48 
60 

72 

4 

8 

12 

5 

10 

15 

20 

6 

12 

18 

24 

30 

36 

42 

7 
8 

14 

16 

21 
24 

28 
32 

35 
40 

42 

48 

49 
56 

56 
64 

63 

72 

70 
80 

77 
88 

84 
96 

9 

18 

27 

36 

45 

54 
60 
66 

63 

70 

77 
84 

72 
80 
88 
96 

81 
^0 
99 

108 

90 
100 
110 
120 

99 
110 
121 
132 

108 
120 
132 
144 

10 

20 

30 

40 

50 

11 

22 

33 

44 

55 

12 

24 

36 

48 

60 

72 

Explanation. — The  numbers  in  the  left-liand  column  may  be 
regarded  as  the  multipliers,  and  the  numbers  across  the  top  as  the 
multiplicands.  The  products  will  be  found  in  the  horizontal  columns 
opposite  the  multipliers. 

Thus,  2  ones  are  2  ;  2  twos  are  4  ;  2  threes  are  6  ;  2  fours  are  8,  etc. 

The  order  may  be  changed  so  that  the  numbers  in  the  upper  hori- 
zontal line  may  be  regarded  as  the  multipliers,  and  the  numbers  on  the 
left  as  the  multiplicands.  Each  successive  multiplier  may  be  used 
with  each  multiplicand. 

Thus,  1  two  is  2 ;  2  twos  are  4 ;  3  twos  are  6 ;  4  twos  are  8,  etc» 


104  ELEMENTS  OF  ARITHMETIC. 

DRILL    EXERCISES. 

Tell  instantly  the  following  products : 

8x4  5x11  12  x6  7x5  2x7 

5x3  12  x7  11  x5  2x6  3x4 

6x    S  9x9  4x9  3x5  2x4 

7x2  8x12  5x5  4x2  12  x3 

9x8  3x3  7x6  5x2  9x4 

8x7  4x10  9x7  6x9  10x12 

3x9  5x7  10  x8  7x11  11  x4 

4x6  6x5  11x10  7x3  2x3 

5x6  11  x7  2x11  2x9  8x6 

3x7  12  x9.  9x2  4x4  6x7 

4x8  7x8  5x9  6x3  12  x8 

5x12  9x6  7x4  10x11  10  x4 

6  X  11  10  X  10  9  X  11  8  X  11  8  X  10 

7x10  11x11  10  X    7  7x12  4x    7 

8x9  12  X  12  8x8  9  X  10  10  x    3 

9x5  7X'7  4x5  10  x6  12  x2 

5x    S  8x5  5  X  10  11  X    9  4  x  12 

3x10  2x8  6x12  12x11  3x2 

4x3  2x12  7x9  2x5  11  x3 

3x11  9x3  6x10  3x8  11  x2 

2  X  10  3  X  12  6  X    6  6x2  11  x    6 

5x4  8x3  12  X    5  10  X    5  12  x  10 

10  X    9  3x6  8x2  11  X  12  10  x    2 

11  X    8  6x4  9  X  12  12  X    4  4  x  11 


MULTIPLICATION.  105 

88.   When  the  multiplier  is  not  greater  than  12. 

ORAL    EXERCISES. 

1.  When  oranges  cost  4  cents  apiece,  how  much  must 
Mary  pay  for  6  oranges  ? 

2.  If  Lulu  changes  6  half-dimes  into  cents,  how  many 
cents  will  she  have  ? 

3.  Henry  went  to  school  every  school-clay  for  8  weekso 
How  many  days  was  that  ? 

4.  A  girl  bought  9  yards  of  ribbon  for  8  cents  a  yard  ? 
How  much  did  it  cost  her  ? 

5.  An  organ-grinder  received  7  cents  at  each  of  8 
houses.     How  much  did  he  get  in  all  ? 

6.  A  boy  sold  9  pencils  at  4  cents  apiece.  How  much 
did  he  get  for  them  all  ? 

7.  In  a  school-room  there  were  8  rows  of  seats,  and  8 
seats  in  a  row.     How  many  pupils  could  be  seated  there  ? 

8.  A  half-quire  of  paper  contains  12  sheets.  How  many 
sheets  are  there  in  6  half-quires  ? 

9.  A  drawing  book  cost  12  cents.  If  each  pupil  in  a 
class  of  9  pupils  paid  the  same  price  for  a  drawing  book, 
how  much  did  they  all  pay  ? 

10.  Harry  had  to  pay  9  cents  per  day  for  his  fare  in 
going  to  and  from  school.     How  much  was  that  per  week  ? 

11.  Alice  planted  9  rows  of  asters  in  her  flower  bed,  and 
each  row  had  9  plants.     How  many  asters  did  she  set  out  ? 

12.  Each  pupil  in  a  class  wrote  7  lines  in  his  writing 
book.  If  there  were  7  pupils  in  the  class,  how  many  lines 
did  they  all  write  ? 

13.  The  cost  of  a  railroad  ticket  from  Ashburn  to  Gray- 
•  stone  is  11  cents.     How  much  will  8  such  tickets  cost  ? 

14.  Some  children  are  required  to  attend  school  6  hours 
per  dayo  How  many  hours  will  they  be  at  school  in  7 
days? 


106  ELEMENTS   OF  ARITHMETIC. 

15.  If  Mary  goes  to  bed  at  8  o'clock  at  night,  and  rises  at 
7  o'clock  in  the  morning,  how  many  hours  does  she  spend 
in  bed  in  9  days  ? 

16.  Eeuben  said  that  he  saw  5  flocks  of  birds,  each  con- 
taining 7  birds.     How  many  birds  did  he  see  ? 

17.  A  school  week  is  5  days.  How  many  days  are  there 
in  9  school  weeks  ? 

18.  Sarah  made  on  her  slate  9  rows  of  squares,  each  row 
containing  7  squares.     How  many  squares  did  she  make  ? 

19.  Henry  brought  home  5  dozen  eggs.  How  many  eggs 
did  he  bring  ? 

20.  Susie's  father  worked  9  hours  per  day  for  7  days. 
How  many  hours  did  he  work  in  all  ? 

21.  How  many  are  two  2's  ?  Three  2's  ?  Four  2's  ? 
Five  2's  ?  etc.     Give  the  table  of  2's. 

22.  Give  the  table  of  3's.  Of  4's.  Of  5's.  Of  6's.  Of 
7's.     OfS's.     Of9's.     OflO's.     Ofll's.     Of  12's. 

23.  Count  by  5's  from  0  to  100  in  this  way :  0,  5,  10,  15, 
20,  25,  30,  etc. 

24.  Count  by  3's  from  0  to  36.     By  Ts  from  0  to  84. 

25.  Count  by  6's  from  0  to  72.     By  9's  from  0  to  108. 

26.  Count  by  4's  from  0  to  48.     By  8's  from  0  to  96. 

27.  Count  by  lO's  from  0  to  110.     By  12's  from  0  to  144. 

WRITTEN    EXERCISES. 
89.   1 .    How  many  are  6  times  397  ? 

Multiplicand,     397         Explanation.  —  For  convenience   in  maltiply- 

- ,  ,,.  ,.  ^     ing,  the  multiplier  is  written  under  the  multiplicand 

Multiplier,  O  »'         ,       .  ,        .   ,  ,  •    i     •     .i     ^  n 

and  we  begm  at  the  right  to  multiply  m  the  follow- 
Product,         Z6'6Z     ing  manner: 

6  times  7  units  are  42  units.  But  42  units  are  4  tens  and  2  units ; 
therefore,  the  figure  2  is  written  in  units'  place  in  the  product,  and  the 
4  tens  are  reserved  to  add  to  the  tens  of  the  product. 

6  times  9  tens  are  54  tens,  plus  4  tens  reserved  are  58  tens,  or  5 
hundreds  and  8  tens ;  therefore  the  figure  8  is  written  in  tens'  place  in 


MULTIPLICATION. 


107 


the  product,  and  the  5  hundreds  are  reserved  to  add  to  the  hundreds 
in  the  product. 

6  times  3  hundreds  are  18  hundreds,  plus  5  hundreds  reserved  are 
23  hundreds,  or  2  thousands  and  3  hundreds,  which  are  written  in 
thousands'  and  hundreds'  places  in  the  product.  Hence  the  product 
is  2382. 


Find  the  products  of  the  following : 


5x46. 
6x37. 
7x28. 
6x93. 
7  x86. 
4x98. 
3x69. 
8x58. 
9  X  43. 
6x85. 

12.  11  X  87. 

13.  12  X  92. 


2 
3 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 


14.  415  X 

15.  372  X 

16.  846  X 

17.  364  X 

18.  918  X 

19.  325  X 

20.  864  X 

21.  325  X 

22.  816  X 

23.  932  X 

24.  519  X  11 

25.  837  X  12 


26.  3245  X 

27.  2861  X 

28.  3942  X 

29.  5687  X 

30.  3834  X 

31.  5279  X 

32.  8168  X 

33.  3934  X 

34.  5468  X 
35  o  3173  X 

36.  4284  X  11 

37.  1987  X  12 


38.  There  are  5280  feet  in  a  mile.     How  many  feet  does 
a  boy  walk  who  walks  5  miles  ? 

39.  A  merchant  sold  Mrs.  Allen  5  pieces  of  muslin,  each 
containing  39  yards.     How  many  yards  did  he  sell  to  her? 

40.  The   average  Avages   of   5   clerks  were   $37.50  per 
month.     How  much  did  they  all  earn  in  a  month  ? 

41.  When  hay  sells  at  $8.25  per  ton,  how  much  must  I 
pay  for  8  tons  ? 

42.  If  a  railway  train  runs  47  miles  per  hour,  how  far 
will  it  run  in  8  hours  ? 

43.  Theodore  Schuyler  &  Co.  paid  each  of  5  bookkeepers 
§85.25  per  month.     How  much  did  they  pay  them  all  ? 

44o    A  laborer  found  that  his  expenses  were  $  23.85  per 
month.     How  much  will  they  be  in  8  months  ? 


108  ELEMENTS  OF  ARITHMETIC. 

45.  A  carriage  maker  sold  8  carriages  for  $78.75  apiece. 
How  much  did  lie  receive  for  them  all  ? 

46.  Mr.  A  loaned  a  man  some  money  and  received  for 
the  use  of  it,  or  interest,  $  18.35  per  year.  How  much 
should  he  receive  for  the  use  of  it  for  5  years  ? 

47.  A  contractor  built  8  houses  for  which  he  received 
$  7325  apiece.     How  much  did  he  receive  for  all  ? 

48.  The  expenses  of  a  large  dry-goods  store  were  $  3215.25 
per  month.     How  much  were  they  for  8  months  ? 

49.  An  organ  manufacturer  sold  during  1891  nine  church 
organs,  the  average  price  of  which  Avas  $  7963.  How  much 
did  his  sales  amount  to  that  year  ? 

50.  The  number  of  papers  sold  by  the  publisher  of  a 
daily  paper  averaged  8432  each  day.  How  many  w^ere  sold 
in  6  days  ? 

51.  A  mechanic  earned  $87.35  per  month  for  8  months. 
How  much  were  his  entire  earnings  in  that  time  ? 

52.  A  man  sold  9  carriages  for  $92.55  apiece.  How 
much  did  he  get  for  them  ? 

53.  The  steamship  Koduna  consumes  125  tons  of  coal 
per  day.  How  much  will  she  consume  in  a  voyage  lasting 
8  days  ? 

54.  In  one  of  the  cities  of  the  United  States,  the  average 
number  of  passengers  carried  upon  the  street-cars  daily  was 
38,395.     How  many  were  carried  in  6  days  ? 

55.  The  sales  of  tickets  at  a  railway  station  for  the 
month  of  January  amounted  to  $38,495.  If  they  are  the 
same,  on  the  average,  for  7  months,  how  much  will  they  be  ? 

56.  The  interest  paid  for  a  sum  of  money  was  $384.27 
per  year.     How  much  interest  money  was  paid  in  5  years  ? 

57.  The  increase  in  population  in  a  city  for  the  year 
1891  was  13,868.  At  that  rate,  what  will  be  the  increase 
in  6  years  ? 

68.    The  average  distance  sailed  by  a  steamship  each  voy- 


MULTIPLICATION.  109 

age  between  New  York  and  Liverpool,  England,  was  2831 
miles o     How  many  miles  did  she  sail  in  4  round  trips  ? 

59.    A  gardener  sold  last  year  9765  bunches  of  asparagus. 
How  many  bunches  would  he  sell  in  7  years  at  that  rate  ? 

Multiply  each  of  the  following  numbers  by  2,  3,  4,  5,  6, 
7,  8,  and  9  : 

60.  5869.     63.  3954.      66.  86,973.      69.  18,572.     72.  48,362. 

61.  7296      64.  3857.      67.  23,547.     70.  42,364.     73.  32,573. 

62.  8164.     65.  5679.      68.  65,132.      71.  38,453.     74.  71,685. 

90.  When  the  multiplier  is  expressed  by  more  than  one  figure. 

ORAL   EXERCISES. 

1.  How  many  are  10  times  6  ?  10  times  7  ?  10  times  8  ? 
10  times  5  ?     10  times  9  ? 

2.  How  many  are  10  times  3  plus  3  times  3  or  13  times 
3?  10  times  4  plus  6  times  4  or  16  times  4?  10  times  7 
plus  3  times  7  or  13  times  7  ?  10  times  8  plus  5  times  8 
or  15  times  8  ? 

3.  How  many  are  15  times  4  ?  18  times  3  ?  17  times  5  ? 
16  times  6  ?     12  times  8  ?     14  times  3  ? 

4.  How  many  are  13  times  5  ?   13  times  6  ?   13  times  7  ? 

5.  What  will  15  yards  of  ribbon  cost  at  5  cents  per 
yard  ?     At  8  cents  per  yard  ? 

6.  How  many  days  are  there  in  17  weeks  ? 

7.  How  many  quarts  are  there  in  16  gallons  ? 

8.  How  many  feet  are  there  in  13  yards  ? 

9.  How  much  will  15  yards  of  calico  cost  at  6  cents  per 
yard  ?  ,  At  9  cents  per  yard  ? 

10.  How  much  does  a  man  save  in  12  weeks  if  he  saves 
^8  per  week? 

11.  There  are  9  square  feet  in  a  square  yard.  How  many 
square  feet  are  there  in  14  square  yards  ? 


110  ELEMENTS  OF  ARITHMETIC. 

12.  How  many  are  10  times  9  ?    10  times  5  ?   10  times  7  ? 

13.  What  figure  must  be  annexed  to  9  to  multiply  it  by 
10?     What  to  5?     What  to  7? 

14.  How,  then,  is  any  number  multiplied  by  10? 

15.  How  many  are  10  sevens  or  10  times  7  ?  100  sevens 
or  100  times  7  ?  '  100  eights  or  100  times  8  ?  100  times  6  ? 
100  times  15?  100  times  35  ?  100  times  48?  100  times  56? 
100  times  84  ?     100  times  97  ?     100  times  63  ? 

16.  How,  then,  is  any  number  multiplied  by  100  ? 

17.  How  many  are  100  times  9  ?  1000  times  9  ?  1000 
times  7  ?  1000  times  15  ?  1000  times  35  ?  1000  times  43  ? 
1000  times  54  ?     1000  times  75  ? 

18.  How,  then,  is  any  number  multiplied  by  1000  ? 

19.  Since  a  number  is  multiplied  by  10  by  annexing  one 
cipher  to  the  number;  by  100  by  annexing  two  ciphers;  by 
1000  by  annexing  three  ciphers,  how  may  a  number  be  mul- 
tiplied by  10,  100,  1000,  etc.,  or  by  1  with  any  number  of 
ciphers  annexed  ? 

91.  A  number  is  multiplied  by  10,  100,  1000,  etc.,  or  by  1 
with  cmy  number  of  ciphers  annexed,  by  annexing  to  the  multi- 
plicand as  many  ciphers  as  there  are  in  the  multiplier. 

WRITTEN    EXERCISES. 

92.  What  are  the  products  of  the  following: 

1.  45  X  10.  6.  324  X  100.  11.  5463  x  1000. 

2.  67  X  10.  7.  416  X  100.  12.  3784  x  1000. 

3.  38  X  10.  8.  723  x  100.  13.  5169  x  10000. 

4.  46  X  10.  9.  342  X  100.  14.  4237  x  10000. 

5.  76  X  10.  10.  518  X  100.  15.  8193  x  100000. 

16.    Multiply  832  x  3000. 

^^^  Explanation.  —Since  3000  is  1000  times  3,  832  is  first 

3000     multiplied  by  8,  giving  2496  as  a  product,  and  this  product 
9J.Qrn00     ^^  multiplied  by  1000  by  annexing  three  ciphers. 


MULTIPLICATION.  m 


17. 

45  X  20. 

27. 

346  X  500. 

37. 

3946  X  6000. 

18. 

37  X  30. 

28. 

279  X  300. 

38. 

5279  X  4000. 

19. 

68  X  40. 

29. 

518  X  700. 

39. 

8365  X  7000. 

20. 

73  X  80. 

30. 

264  X  900. 

40. 

7226  X  9000. 

21. 

56  X  60. 

31. 

385  X  400. 

41. 

5387  X  8000. 

22. 

90  X  40. 

32. 

498  X  600. 

42. 

7394  X  5000. 

23. 

30  X  50. 

33. 

341  X  800. 

43. 

2869  X  3000. 

24. 

23  X  90. 

34. 

957  X  200. 

44. 

9999  X  2000. 

25. 

84  X  70. 

35. 

691  X  500. 

45. 

8765  X  4000. 

26. 

65  X  60. 

36. 

444  X  400. 

46. 

9182  X  6000. 

47. 

Multiply 

373 

X  235. 

First  Process. 

373  Second  Process. 

373 

235 

235 

Ist  partial  product,      1865  1865 

2d  partial  product,     11190  1119 

3d  partial  product,     74600  746 


Entire  product,  87655  87655 

Explanation  of  First  Process,  —  For  convenience,  the  multiplier 
is  written  under  the  multiplicand  with  units  under  units,  tens  under 
tens,  etc. 

Since  we  cannot  multiply  by  235  at  one  operation,  we  multiply  by 
the  parts  of  235,  viz.  :  5  units,  3  tens  or  30,  and  2  hundreds  or  200, 
and  add  the  products. 

5  times  373  is  1865,  the  first  partial  product ;  30  times  373  is  11190, 
the  second  partial  product ;  200  times  373  is  74600,  the  third  partial 
product.  The  sum  of  these  partial  products  is  87655,  the  entire 
product. 

Explanation  op  Second  Process.  —  The  second  process  is  the  one 
commonly  used.  It  is  the  same  as  the  first  except  that  the  ciphers  are 
omitted  from  the  right  of  the  partial  products.  The  significant  figures, 
however,  occupy  their  proper  places. 

Thus,  in  multiplying  by  3  tens  the  product  is  1119  tens,  and  conse- 
quently, the  right-hand  figure  of  this  product  is  placed  under  tens  and 
the  others  in  their  proper  places  in  the  product. 

In  multiplying  by  2  hundreds  or  any  number  of  hundreds  the  lowest 
order  of  the  product  is  hundreds,  consequently  the  right-hand  figure 
of  the  product  is  written  under  hundreds. 


112 


ELEMENTS   OF   ARITHMETIC. 


93.  From  these  processes  the  following  general  method  of 
multiplying  i^  readily  deduced : 

1.  The  multiplier  is  imntten  under  the  multiplicand,  units 
under  units,  tens  under  tens,  etc. 

2.  Each  figure  of  the  multiplicand  is  multiplied  by  each 
signijicant  figure  of  the  midtiplier  and  the  right-hand  figure  of 
each  product  is  placed  under  the  figure  of  the  multiplier  used 
to  obtain  it. 

3.  The  sum  of  the  p)artial  products  luill  be  the  entire  product. 

When  there  is  a  cipher  in  the  multiplier,  multiply  by  the  significant 
figures  only,  taking  care  to  place  the  right-hand  figure  of  each  partial 
product  under  the  figure  used  to  obtain  it. 


48. 

49. 

50. 

5684 

78645 

59658 

609 

5008 

3700 

51156 

629160 

417606 

34104 

393225 

1/ 

'8974 

3461556 

393854160 

220734600 

■  Multiply : 

51. 

267  by  24. 

66. 

3845  by  236. 

81. 

5698  by  792. 

52. 

836  by  27. 

67. 

7186  by  314. 

82. 

3179  by  865. 

53. 

918  by  61. 

68. 

5327  by  420. 

83. 

8169  by  734. 

54. 

309  by  38. 

69. 

6284  by  518. 

84. 

5234  by  826. 

55. 

465  by  27. 

70. 

4769  by  344. 

85. 

3186  by  839. 

56. 

732  by  55. 

71. 

6873  by  530. 

86. 

5279  by  918. 

57. 

817  by  63. 

72. 

2918  by  364. 

87. 

3864  by  609. 

58. 

398  by  84. 

73. 

4873  by  273. 

88. 

38,675  by  708. 

59. 

426  by  29. 

74. 

9275  by  731. 

89. 

42,308  by  692. 

60. 

918  by  76. 

75. 

8462  by  820. 

90. 

75,069  by  804. 

61. 

837  by  84. 

76. 

7319  by  394. 

91. 

83,427  by  3625- 

62. 

695  by  32. 

77. 

8425  by  972. 

92. 

80,069  by  5846. 

63. 

864  by  69. 

78. 

6685  by  873. 

93. 

58,325  by  3192. 

64. 

528  by  87. 

79. 

6517  by  762. 

94. 

38,164  by  8068. 

65. 

364  by  44. 

80. 

7395  by  469. 

95. 

49,234  by  9096. 

MULTIPLICATION.  113 

96.  f  48.27  by  584.  101.  $543.79  by  3980. 

97.  $39.35  by  376.  102.  $  708.27  by  2900. 

98.  $  73.86  by  705.  103.  $  496.83  by  5805. 

99.  $85.18  by  835.  104.  $  385.76  by  3760. 
100.  $49.66  by  597.  105.  $483.74  by  5736. 

106.  A  manufacturer  sold  29  reapers  for  $  96.85  ajoiece. 
How  much  did  he  get  for  them  ? 

107.  A  man  walked  37  miles.  How  many  feet  did  he 
walk,  since  there  are  5280  feet  in  a  mile  ? 

108.  A  tract  of  land  containing  294  acres  was  sold  for 
$  313  per  acre.     For  ho^v  much  did  it  sell  ? 

109.  It  required  318  loads  of  bricks  averaging  1387  bricks 
to  build  a  school  building.     How  many  bricks  was  that  ? 

110.  A  railroad  employed  213  switchmen,  whose  wages 
averaged  $  78.34  per  month.  How  much  did  all  the  switch- 
men receive  per  month  ? 

111.  A  merchant  sold  7  pieces  of  cloth,  each  containing 
34  yards,  for  $2.17  per  yard.  How  much  did  he  receive 
for  it  ? 

112.  The  Danton  Leather  Co.  purchased  37  lots  of  hides, 
each  lot  containing  384  hides.  How  many  did  they  pur- 
chase ? 

113.  A  cloth  dealer  sold  54  pieces  of  cassimere,  each  con- 
taining 48  yards,  at  $  1.87  per  yard.  How  much  did  he 
receive  for  it  ? 

114.  A  fire  destroyed  345  gross  of  pencils.  Since  a  gross 
is  144,  how  many  pencils  were  destroyed  ? 

115.  In  a  day  there  are  1440  minutes.  How  many  min- 
utes are  there  in  a  year  ? 

116.  A  farm  of  95  acres  was  sold  for  $37.25  per  acre. 
How  much  was  received  for  it  ? 

117.  If  your  pulse  beats  4320  times  per  hour,  how  many 
times  will  it  beat  in  2  days  of  24  hours  each  ? 

118.  The  cost  of  building  a  railroad  was  $  31,755  per  mile. 

EL.    OF    AU.  —  8 


114  ELEMENTS  OF  ARITHMETIC. 

If  it  was  118  miles  long^  how  much,  did  it  cost  to  construct 
it? 

119.  A  pump  can  pump  365,864  gallons  per  day.  How 
many  gallons  can  it  pump  in  31  days  ? 

120.  The  expense  of  an  army  in  Europe  is,  on  the  average, 
$  195  per  man.  How  much  will  it  cost  to  keep  an  army  of 
32,596  men  ? 

121.  If  a  locomotive  costs  $19,325,  how  much  will  23 
such  locomotives  cost  ? 

122.  An  army  used  3285  barrels  of  flour  each  day  for  49 
days.     How  many  barrels  did  it  use  altogether  ? 

123.  A  shoe  manufacturer  in  Massachusetts  sold  23,876 
pairs  of  shoes  at  an  average  price  of  $3.15  a  pair.  How 
much  did  he  receive  for  them  ? 


REVIEW. 

ORAL  EXERCISES. 

94.  1.  Henry  bought  3  pencils  at  3  cents  apiece,  and  5 
postage  stamps  at  2  cents  apiece.  How  much  did  he  pay 
for  all  ? 

2.  The  wages  of  a  clerk  were  $  15  per  week.  If  he  paid 
$  5  for  board  and  $  4  for  other  expenses,  how  much  did  he 
save  in  5  weeks  ? 

3.  Nina  spent  20  cents  for  a  box  of  water-color  paints,  5 
cents  for  brushes,  and  had  10  cents  left.  How  much  money 
had  she  at  first  ? 

4.  Two  men  travel  in  opposite  directions  from  a  starting- 
point,  one  at  the  rate  of  5  miles  per  hour,  and  the  other  at 
the  rate  of  4  miles  per  hour.  How  far  apart  are  they  at 
the  end  of  5  hours  ? 

5.  Mary  counted  the  petals  on  6  flowers.  All  but  2  of 
them  had  6  petals  each,  and  the  others  had  5  each.  How 
many  petals  had  they  all  ? 


MULTIPLICATION.  115 

6.  Hannah  was  at  school  6  hours  per  day  for  4  days,  and 
3  hours  on  the  other  day.  How  many  hours  was  she  at 
school  that  week  ? 

7.  Warren  had  5  hens  and  each  hen  had  7  chickens. 
How  many  chickens  were  there  ? 

8.  A  grocer  sold  5  pounds  of  sugar  at  5  cents  a  pound, 
and  received  in  payment  a  fifty-cent  piece.  How  much 
change  should  he  give  back. 

9.  Henry  counted  a  company  of  soldiers  in  this  way: 
he  found  that  there  were  8  rows  of  9  men  each  and  three 
officers.     How  many  soldiers  did  he  see  ? 

10.  Some  children  found  5  birds'  nests  containing  5  eggs 
each,  and  4  nests  containing  4  eggs  each.  How  many 
eggs  did  they  find  ? 

11.  A  farmer  sold  5  calves  at  $  5  each,  and  5  sheep  at 
$  4  each.     How  much  did  he  receive  for  all  ? 

12.  Mary  paid  10  cents  for  a  dozen  peaches,  and  2  cents 
apiece  for  a  dozen  bananas.  How  much  did  they  all  cost 
her? 

13.  A  boy  worked  6  days  for  5  dimes  per  day,  but  he 
spent  2  half-dimes  for  car  fare  each  day.  How  much  did 
he  save  ? 

14.  Two  brothers  worked  during  the  summer  vacation  of 
10  weeks,  one  for  f  6  per  week  and  the  other  for  $  3  per 
week.  How  much  more  did  the  one  earn  than  the  other  in 
that  time  ? 

15.  The  pupils  in  a  school  were  divided  into  8  classes  or 
grades,  each  containing  9  pupils.  If  every  class  contained 
5  girls  and  4  boys,  how  many  boys  were  there  in  the  school  ? 
How  many  girls  ?     How  many  pupils  ? 

16.  A  merchant  employed  5  clerks  at  $8  per  week,  and 
2  at  $  10  per  week.  How  much  did  he  pay  them  all  per 
week? 

17.  The  retail  price  of  asparagus  was  15  cents  a  bunch, 


116  ELEMENTS  OF   ARITHMETIC. 

and  the  wholesale  price  was  12  cents.  How  much  is  a 
retail  dealer's  profit  on  20  bunches  which  he  bought  at 
wholesale  ? 

18.  A  hatter  bought  12  hats  at  $4  apiece,  and  sold  them 
at  $  6  apiece.     How  much  did  he  gain  ? 

19.  A  clothier  sold  overcoats  at  $  12.50  which  cost  him 
$  9.50.     How  much  was  his  gain  on  20  coats  ? 

20.  After  paying  ^2  apiece  for  repairing  clocks  that 
cost  $  10  apiece,  a  man  sold  12  of  them  for  $  144.  Did  he 
gain  or  lose  by  the  sale,  and  how  much  ? 

WRITTEN     EXERCISES. 

95.  1.  A  bookkeeper  received  $75  per  month  for  his 
services,  but  he  spent  in  a  year  f  680.  How  much  did  he 
save  in  a  year  ? 

2.  A  train  composed  of  23  cars,  containing  145  sheep 
each,  arrived  at  Boston  and  the  sheep  were  sold  at  $  5  a  head. 
How  much  was  received  for  them  ? 

3.  A  boy  had  $350  when  he  was  15  years  old,  and  he 
saved  $  100  per  year  until  he  was  21.  How  much  had  he 
then  ? 

4.  A  manufacturer  employed  118  workmen  whose  aver- 
age wages  per  month  were  $  68.  How  much  did  they  all 
earn  per  month  ? 

5.  A  drover  bought  83  cattle  at  $37.25  per  head.  He 
sold  50  of  them  at  $45  per  head,  and  the  rest  at  $35.25  per 
head.     How  much  did  he  gain  ? 

-  6.  A  tea  merchant  sold  348  pounds  of  tea  at  57  cents 
per  pound  and  gained  $  13.55.  How  much  did  it  cost  him  ? 
7.  A  and  B  each  began  business  with  $4000.  A  gained 
$600  the  first  year  and  B  lost  an  equal  sum.  The  second 
year  A  gained  $  318  and  B  gained  $  895.  How  much  was 
each  worth  then  ? 


MULTIPLICATION.  117 

8.  A  farmer  raised  895  bushels  of  oats.  He  kejjt  50 
bushels  for  seed,  and  enough  to  keep  10  horses,  allowing  15 
bushels  for  each  horse,  and  sold  the  rest  at  35  cents  per 
bushel.     How  much  did  he  receive  for  the  amount  sold  ? 

9.  A  coal  dealer  sold  315  tons  of  coal  at  $5.25  per  ton. 
He  received  in  payment  a  note  for  $  1000  and  the  rest  in 
cash.     How  much  cash  did  he  receive  ? 

10.  The  profits  of  a  merchant  were  $  5896.75.  He  paid 
$  1391  for  his  household  expenses,  and  3  times  as  much  for 
other  expenses.     What  was  his  net  gain? 

11.  What  will  be  the  cost  of  485  chests  of  tea,  each  con- 
taining 55  pounds,  at  38  cents  a  pound  ? 

12.  I  had  a  farm  of  313  acres.  185  acres  yielded  me  an 
income  of  $  6.25  per  acre,  and  the  rest  $  6  per  acre.  How 
much  was  my  entire  income  from  the  farm  ? 

13.  My  agent  sold  3  lots  of  goods.  For  the  first  he 
received  $  945,  for  the  second  twice  as  much  as  for  the  first, 
and  for  the  third  3  times  as  much  as  for  the  other  two. 
How  much  did  he  receive  for  all  ? 

14o  Three  men  formed  a  partnership.  A  contributed 
$690,  B  twice  as  much,  and  C  three  times  as  much  as  A  and 
B.     What  was  their  entire  capital  ? 

15.  A  drover  bought  132  cattle  at  $45  per  head,  and  67 
at  $  61  per  head.  He  sold  them  all  at  $  50  per  head.  Did 
he  gain  or  lose,  and  how  much  ? 

16.  A  farmer  sold  his  farm  at  a  gain  of  $  1685.  The  barns 
cost  him  $  1855,  his  house  $  8520,  and  the  land  twice  as  much 
as  the  barns  and  house.     How  much  did  he  get  for  the  farm  ? 

17.  A  merchant  bought  35  pieces  of  carpet,  each  con- 
taining 49  yards,  at  $1.18  per  yard.  He  paid  cash  for  a 
part  of  the  cost  and  a  note  to  settle  the  account  for  $  1250. 
How  much  cash  did  he  pay  ? 

18.  Three  men  began  business  at  the  same  time.  During 
the  first  year  A  made  a  profit  of  $  S55,  B  of  3  times  as 


118  ELEMENTS  OF  ARITHMETIC. 

niuclij  and  C  of  4  times  as  much  as  A.  At  the  beginning 
of  the  second  year  they  formed  a  partnership  in  which  each 
man  contributed  his  gain  of  the  previous  year,.  What  was 
the  entire  capital  ? 

19.  The  daily  sales  cf  a  newspaper  for  5  days  were 
16,845  copies.  If  the  papers  were  sold  for  3  cents  each, 
what  was  the  amount  received  ? 

20.  I  know  a  man  whose  income  is  $  85.75  per  day.  How 
much  does  he  receive  in  a  year,  or  365  days  ? 

21.  The  annual  expense  of  a  school  for  5  years  was 
$  3825.25,  exclusive  of  the  wages  of  the  teachers.  In  the 
school  there  were  11  teachers  whose  average  annual  salary 
was  $  815.S5.    What  was  the  annual  expense  of  the  school  ? 

22.  It  was  discovered  in  taking  a  census  that  a  certain 
town  contained  145  houses,  that  each  house,  on  the  average, 
contained  2  families,  and  that  each  family  averaged  6  per- 
sons.    How  many  people  were  there  in  the  town  ? 

23.  A  man  bought  a  farm  of  189  acres.  He  sold  99 
acres  at  $  87  per  acre,  and  thus  reduced  the  cost  of  the  rest 
to  $  58  per  acre.     How  much  did  the  farm  cost  him  ? 

24.  A  single  poppy  has  been  known  to  contain  32,000 
seeds.     How  many  seeds  will  335  such  poppies  contain  ? 

25 o  In  a  codfish  9,334,000  eggs  have  been  found.  If 
from  each  egg  a  fish  was  hatched,  how  many  fishes  were 
hatched  from  the  eggs  of  27  such  codfish  ? 

26.  A  man  set  out  on  foot  to  travel  600  miles.  He 
traveled  during  the  first  5  days  45  miles  per  day ;  during 
the  next  6  days  38  miles  per  day.  How  many  miles  had 
he  still  to  travel  ? 

27.  A  ship  left  port,  sailing  at  the  rate  of  9  miles  per 
hour.  After  sailing  for  23  hours  she  was  driven  directly 
back  by  a  storm  at  the  rate  of  12  miles  per  hour  for  10 
hours.  How  far  v/as  she  from  the  port  from  which  she  sailed, 
at  the  end  of  30  hours  ? 


DIVISION. 


96.  1.  How  many  groups  of  2  squares  can  be  formed 
from  6  squares  ?     How  many  2's  are  there  in  6  ? 

2.  How  many  groups  of  3  oranges  can  be  formed  from  9 
oranges  ?     How  many  3's  are  there  in  9  ? 

3.  Henry  earns  6  cents  an  hour.  In  how  many  hours 
can  he  earn  42  cents  ?     How  many  6's  are  there  in  42  ? 

4.  Louise  arranged  56  spools  in  rows,  placing  7  in  each 
row.     How  many  rows  did  she  form  ? 

5  c  Daisy's  brother  was  ill  for  49  days.  How  many  weeks 
was  he  ill  ?     How  many  7's  are  there  in  49  ? 

6.  Anna  had  63  chickens  which  were  hatched  in  broods 
of  9  chickens  each.  How  many  broods  were  there  ?  How 
many  9's  are  there  in  63  ? 

7.  A  lad  rides  each  day  8  miles  in  going  to  and  from 
school.  In  how  many  days  will  he  ride  96  miles  ^  How 
many  times  is  8  contained  in  96  ? 

8.  Mollie  arranged  12  books  in  4  piles.  How  many 
books  were  there  in  each  pile  ? 

9.  How  many  apples  will  each  child  receive,  when  10 
apples  are  divided  equally  among  5  children  ? 

10.    What  have  you  been  doing  with  the  above  numbers  ? 

97.  The  process  of  finding  how  many  times  one  number 
is  contained  in  another,  or  the  process  of  separating  a  num- 
ber into  equal  parts,  is  called  Division. 

98.  The  number  to  be  divided  is  the  Dividend. 

119 


120  ELEMENTS   OF   ARITHMETIC. 

99.   The  number  by  wbicli  we  divide  is  the  Divisor. 

100.  The  result  obtained  by  division  is  the  Quotient. 

101.  The  part  of  the  dividend  remaining  when  the 
division  is  not  exact,  is  the  Remainder. 

102.  The  Sign  of  Division  is  -4-.  It  is  read  divided  by. 
When  placed  between  two  numbers,  it  shows  that  the  one 
at  the  left  is  to  be  divided  by  the  one  at  the  right. 

Thus,  24  -T-  6  is  read,  24  divided  by  6. 

103.  Division  is  also  indicated  by  writing  the  dividend 
above  the  divisor,  with  a  line  between  them. 

Tims,  '^f-  is  read,  24  divided  by  6. 

Another  method  of  indicating  division  is  by  writing  the 
divisor  at  the  left  of  the  dividend,  with  a  line  between  them. 
Thus,  6)24  is  read,  24  divided  by  6. 

DRILL  EXERCISES. 

104.  Give  the  results  instantly : 


9x 

2  =  ? 

9x  ? 

=  18 

? 

X 

2 

=  18 

18 

—  2  =  ? 

6x7  =  ? 

6x? 

=  42 

? 

X 

7 

=  42 

42 

—  7  =  ? 

5x8  =  ? 

5x? 

=  40 

? 

X 

8 

=  40 

40 

-8  =  ? 

7x5=? 

7x  ? 

=  35 

? 

X5 

=  35 

35 

-5  =  ? 

8x3  =  ? 

8x? 

=  24 

9 

X 

3 

=  24 

24 

-3  =  ? 

4x9=? 

4x  ? 

=  36 

? 

X 

9 

=  36 

36 

-9=? 

6x8  =  ? 

6  X  ? 

=  48 

? 

X 

8 

=  48 

48 

-v-8=? 

20- 

-10 

21 

-^    7 

48- 

12 

50- 

10 

22—2 

8- 

-   4 

48 

^   4 

24- 

4 

27- 

9 

32-8 

20- 

-    2 

42 

-f-   6 

16- 

2 

66- 

6 

60-   5 

96- 

-   8 

44 

-11 

24- 

6 

60- 

12 

88-11 

6- 

-   3 

28 

-    4 

30- 

5 

63- 

7 

40-5 

22- 

-11 

36 

-    4 

m^ 

11 

48- 

8 

44-4 

12- 

:-     2 

70 

-10 

8- 

2 

49- 

7 

96  - 12 

20- 

-    4 

63 

-    9 

81 -i- 

9 

35  -j- 

7 

24-3 

DIVISIOX. 

121 

24-12 

88-    8 

35-5 

.    90-10 

45-9 

18-    3 

55-11 

80-10 

48-6 

108-    9 

60-6 

84-7 

10-2 

110-11 

121  -  11 

15-3 

16-4 

77  - 11 

80-   8 

108  -  12 

56-7 

14-7 

6-    2 

99-9 

21-3 

90-9 

4--    2 

36-   9 

40-8 

132  -  12 

70-7 

16-8 

84-12 

50-5 

120  -  10 

72-    8 

72-12 

18-2 

27-3 

110-10 

20-5 

30-3 

54-9 

100  -  10 

120  -  12 

12-6 

18-    6 

12-3 

24-8 

132  -  11 

33-11 

10-5 

12-4 

9-    3 

144  -  12 

40-4 

15-5 

33-3 

36-6 

42-7 

25-5 

30-10 

32-4 

77-7 

60-10 

30-6 

24-2 

40-10 

99-11 

56-8 

18-9 

72-6 

64-8 

28-7 

36-3 

54-6 

36  - 12 

55-5 

72-    9 

14-    2 

105.   When  the  divisor  is  not  greater  than  i2„ 

ORAL    EXERCISES. 

1.  When  coal  is  selling  at  $  5  per  ton,  how  many  tons 
can  be  bought  for  $  45  ? 

2.  How  many  rows  of  trees  are  there  in  an  orchard  con- 
taining 60  trees  arranged  in  rows  containing  6  trees  each  ? 

3.  If  a  railroad  ticket  to  Chicago  costs  $  7,  how  many 
such  tickets  can  be  bought  for  $35? 

4.  A  girl  paid  96  cents  for  muslin  at  8  cents  per  yard. 
How  many  yards  did  she  buy  ? 

5.  How  many  dictionaries,  at  $S  apiece,  can  be  bought 
for  $U? 

6.  How  long  will  it  take  a  boy  to  save  §48  if  he  saves 
f  4  per  month  ? 

7.  Howard  paid  $9  for  a  suit  of  clothes.     How  many 
such  suits  can  be  bought  for  $  54  ? 

8.  How  many  gallons  of  molasses  are  there  in  44  quarts  ? 


122  ELEMENTS  OF  ARITHMETIC. 

9.    A  peck  measure  holds  8  '][uarts  of  grain.     How  many 
pecks  are  there  in  72  quarts  ? 

10.  If  8  nails  are  required  to  fasten  a  horseshoe,  how 
many  horseshoes  can  be  fastened  with  80  nails  ? 

11.  How  many  pairs  of  boots,  at  $  6  a  pair,  can  be  bought 
for  $48? 

12.  A  farmer  had  63  bales  of  hay.  If  he  drew  it  away 
by  drawing  7  bales  at  a  load,  how  many  loads  did  it  make  ? 

13.  A  grocer  put  96  quarts  of  vinegar  into  two-gallon 
jugs.     How  many  jugs  did  he  use  ? 

14.  A  stick  measured  72  inches  in  length.  How  many 
feet  long  was  it  ? 

15.  Henry  found  that  he  had  added  66  figures,  and  that 
there  were  6  figures  in  each  column.  How  many  columns 
did  he  add? 

16.  Mary  knew  that  there  were  72  desks  in  her  school- 
room, and  that  there  were  8  desks  in  a  row,  but  she  could 
not  tell  how  many  rows  there  were.     Can  you  tell? 

17.  A  grocer  had  76  eggs.  How  many  dozen  did  he 
have,  and  how  many  over  ? 

18.  A  lad  saw  48  birds  sitting  upon  6  telegraph  wires, 
and  he  noticed  that  the  same  number  sat  upon  each  wire. 
How  many  were  there  in  each  group  ? 

19.  Lottie  had  saved  47  cents  Avhich  she  wished  to  divide 
equally,  as  far  as  possible,  among  four  charitable  societies. 
How  much  did  she  give  each,  and  how  much  had  she  left  ? 

20.  How  many  times  can  a  four-quart  measure  be  filled 
from  49  quarts  ?     How  much  will  be  left  ? 

21.  A  postman  found  that  he  walked  24  miles  per  day  in 
delivering  mail.  If  he  made  4  trips  per  day,  how  many 
miles  did  he  walk  each  trip  ? 

22.  Susie  spends  -^  of  the  year  with  her  aunt  and  the 
rest  of  the  year  at  home.  How  many  months  does  she 
spend  with  her  aunt  ? 


DIVISION.  123 


WRITTEN   EXERCISES. 

106.  1.    Divide  1702  by  3. 

Divisor.  Dividend,  Explanation.  —  For     convenience,    the 

3)1702  divisor  is  written  at  the  left  of  the  divi- 

37^      ^  dend,  and  the  quotient  below  it  or  above  it. 

bbi  —  i,  Rem.  ^^^  ^^^.^  ^^  ^^^^  ^^^^  ^^  divide.     Thus,  3 

^^°^^^°  ■  is  not  contained  in  1  thousand  any  thousand 

times,  therefore  the  quotient  cannot  contain  units  of  any  order  higher 
than  hundreds.  Hence,  we  find  how  many  times  3  is  contained  in  all 
the  hundreds  of  the  dividend. 

1  thousand  plus  7  hundreds  are  17  hundreds  ;  3  is  contained  in  17 
hundreds  5  hundred  times  and  2  hundreds  remainder.  The  5  is  writ- 
ten in  hundreds'  place  in  the  quotient,  and  the  remainder,  2  hundreds, 
is  united  with  the  tens. 

2  hundreds  plus  0  tens  are  20  tens  ;  3  is  contained  in  20  tens  6 
times,  and  a  remainder  of  2  tens.  The  6  is  written  in  tens'  place  in 
the  quotient,  and  the  remainder,  2  tens,  is  united  with  the  units. 

2  tens  plus  2  units  are  22  units  ;  3  is  contained  in  22  units  7  times, 
and  a  remainder  of  1.  The  7  is  written  in  units'  place  in  the  quotient, 
and  the  remainder  at  a  little  distance  to  the  right ;  or  it  may  be  written 
as  a  part  of  the  quotient  by  writing  the  divisor  under  it. 

Hence,  the  quotient  is  567  and  1  remainder,  or  567A. 

Proof.  —  567,  the  quotient,  multiplied  by  3,  the  divisor,  plus  1,  the 
remainder,  is  equal  to  1702,  the  dividend.     Hence,  the  work  is  correct. 

107.  When  examples  in  division  are  solved  without 
writing  the  products  or  remainders,  the  process  is  called 
Short  Division. 

Short  division  is  not  generally  employed  when  the  divisor  exceeds  12. 

Divide  by  short  division,  and  prove : 

4.  5. 

7)59273     8)614653 
84674       76831f 

3936-4.  16.  $1125-9. 

3234-3.  17.  31477-7. 

5256^6.  18.  $4986-9. 

1362-3.  19.  $6765-5. 

1328-8.  20.  $3836-7. 


2. 

3. 

4)3624 

6)65814 

906 

10969 

6.    4872-4. 

11.    , 

7     (3830  -  5. 

12.  ; 

8.    2970-5. 

13.    , 

9.    2976-6. 

14.  : 

10.   4635-3. 

15.  : 

124  ELEMENTS   OF   ARITHMETIC. 

21.  9872 -f- 6.  28.      $9185-f-    4.  35.    i-^/^. 

22.  4567-5.  29.    $  84364 -^    7.  36.    i-V-- 

23.  $5  89.32 --6.  30.    1 38857 -J-    8,  37.    ^7_37- 

24.  $  81.74 -f- 9.  31.    ^  56878 -f-    6.  38.    ^\^-^. 

25.  $83424 --8.  32.    $91341--    9.  39.    Aie^s. 

26.  $73265 --5.  33.    $72144--    6.  40.    ^f^i. 

27.  $58624 --7.  34.    $31486 --11.  41.    5  2_6^^ 

42.  How  many  coats,  at  $5  each,  can  be  bought  for 
$  2250  ? 

43.  If  a  man  spends  $  7  per  week,  how  long  will  it  take 
him  to  spend  $  1477  ? 

44.  The  salaries  of  5  men  amounted  to  $3765.  What 
was  the  average  salary  ? 

45.  If  a  ship  sails  8  miles  per  hour,  how  long  will  it  take 
her  to  sail  1248  miles  ? 

46.  A  stage  traveled  6  miles  per  hour.  How  many  hours 
would  it  require  to  travel  2352  miles  ? 

47.  At  a  church  collection,  each  person  contributed,  on 
the  average,  7  cents.  If  the  entire  sum  collected  was  4823 
cents,  how  many  persons  contributed  ? 

48.  How  many  yards  of  cloth,  at  9  cents  per  yard,  can  be 
bought  for  29,439  cents  ? 

49.  How  many  cloaks,  at  $5  each,  can  be  bought  for 
$32,615? 

50.  In  measuring  the  depth  of  the  sea,  6  feet  are  called 
a  fathom.  How  many  fathoms  deep  is  the  sea  when  it 
measures  3516  feet  ? 

Divide  each  of  the  following  numbers  by  3,  4,  5,  6,  7,  8, 
and  9. 


51. 

1950. 

55. 

5616. 

59. 

17286. 

63. 

277223. 

52. 

1476. 

56. 

6909. 

60. 

11615. 

64. 

548337. 

53. 

4360. 

57. 

2616. 

61. 

44854. 

65. 

389648. 

54. 

2943. 

58. 

1884. 

62. 

18809. 

66. 

261950. 

DIVISION.  125 

108.  When  the  divisor  is  expressed  by  more  than  one  figure. 

ORAL    EXERCISES. 

1.  At  10  cents  a  yard,  how  many  yards  of  cambric  can 
be  bought  for  40  cents  ? 

2.  Tliere  are  25  cents  in  a  quarter  of  a  dollar.  To  how 
many  quarter  dollars  are  50  cents  equal  ? 

3.  If  I  pay  $  60  for  20  lambs,  how  much  do  I  pay  for  each? 

4.  A  real  estate  dealer  divided  30  acres  into  10  equal 
lots.     How  many  acres  were  there  in  each  ? 

5.  If  a  newsboy  sells  40  papers  per  day,  in  how  many 
days  will  he  sell  120  papers  ? 

6.  If  I  sell  books  at  30  cents  each,  how  many  must  I 
sell  to  receive  90  cents  ? 

7.  How  many  lO's  are  there  in  60?  In  90?  In  120? 
In  250?     In  350?     In  480  ? 

8.  Since  in  60  there  are  6  tens ;  in  90,  9  tens ;  in  120, 
12  tens ;  in  250,  25  tens,  how  may  any  number  be  divided 
by  10  ? 

9.  How  many  lOO's  are  there  in  600  ?  In  900  ?  In 
1200  ?     In  3600  ?     In  7500  ?     In  8400  ? 

10.  Since  in  600  there  are  6  hundreds ;  in  900,  9  hun- 
dreds ;  in  1200,  12  hundreds ;  in  3600,  36  hundreds,  how 
ma}^  any  number  be  divided  by  100  ? 

11.  How  many  lOOO's  are  there  in  8000?     In  25000  ? 

12.  Since  in  8000  there  are  8  thousands,  in  25000  there 
are  25  thousands,  how  may  a  number  be  conveniently 
divided  by  1000? 

How  may  a  number  be  divided  by  10,  100,  1000,  or  by  1 
with  any  number  of  ciphers  annexed? 

109.  A  number  may  be  divided  by  10,  100,  1000,  or  by  1 
ivith  any  number  of  ciphers  annexed,  by  cutting  off  from  the 
right  of  the  dividend  as  many  figures  as  there  are  ciphers  on 
the  right  of  the  divisor. 


126 


ELEMENTS   OF  ARITHMETIC. 


WRITTEN    EXERCISES. 


110.  Divide: 

1.  1|0)3968|0 

2.  1|00)382|00 

3.  1|00)468|57 

3968 

382 

468^^ 

4. 

3720  by  10. 

9. 

57386  by  100. 

14. 

5283a  by  1000. 

6. 

4860  by  10. 

10. 

42570  by  100. 

15. 

32685  by  1000. 

6. 

3879  by  10. 

11. 

29830  by  100. 

16. 

97237  by  1000. 

7. 

4265  by  10. 

12. 

34206  by  100. 

17. 

54862  by  1000. 

8. 

3842  by  10. 

13. 

51309  by  100. 

18. 

69154  by  1000. 

19.    Divide  38476  by  500. 

Explanation, — The  number  is  divided  by  100 
by   cutting  off  the   two  right-hand  figures  of  the 


5[00)384|76 


'^^lo-f       dividend.     Dividing  384  by  5  is,  therefore,  dividing 

38,400  by  500,     The-  quotient  is,  therefore,  76  and 

4  hundreds  remainder.     This  remainder,    added  to  the  remainder 

obtained  by  dividing  by  100,  gives  the  entire  remainder,  476 ;  or  the 

quotient  is  76|^f . 

Divide : 

20.  6835  by  20. 

21.  7291  by  30. 

22.  4863  by  60. 

23.  3925  by  80. 


24.  98746  by  300. 

25.  38295  by  400. 

26.  41283  by  700. 

27.  61579  by  900. 
32.  Divide  15989  by  37. 

Divisor.   Dividend.  Quotient.  EXPLANATION. 

37)  15989  (432-jV 
148 


28.  83426  by  500. 

29.  31284  by  800. 

30.  67395  by  700. 

31.  81293  by  600. 


118 
111 


When  the  divisor  is 
larger  tlian  12  the  steps  in  the  sokition  are 
usually  w^ritten.  Thus,  37  is  contained  in 
159  four  times  and  a  remainder  of  11.  The 
4  is  v^ritten  in  the  quotient  and  the  next 
figure  of  the  dividend  annexed  to  the  re- 
mainder, making  118. 

37  is  contained  in  118  three  times  and  a 
remainder  of  7.     The  3  is  written  in  the 
quotient  and  the  next  figure  of  the  dividend 
annexed  to  the  remainder,  making  70. 

37  is  contained  in  79  two  times  and  a  remainder  of  5.     Therefore, 
the  quotient  is  432  g^. 


79 
74 

5 


DIVISION.  127 

When  tlie  steps  in  the  solution  of  an  example  are  written, 
the  process  is  called  Long  Division. 

111.  From  the  solution  of  the  example  given,  the  follow 
ing  are  seen  to  be  the  successive  steps  taken  in  solving  an 
example  in  Long  Division  : 

Write  the  divisor  at  the  left  of  the  dividend,  ivith  a  curved 
line  hetiueen  them. 

Find  how  many  times  the  divisor  is  contained  in  the  fewest 
figures  07i  the  left  hand  of  the  dividend  that  ivill  contain  it,  and 
write  the  quotient  at  the  right  of  the  dividend  or  above  it. 
Multiply  the  divisor  by  this  quotient,  and  place  the  product 
under  the  figures  divided.  Subtract  the  result  from  the  par- 
tial dividend  used,  and  to  the  remainder  annex  the  next  figure 
of  the  dividend. 

Divide  as  before  until  all  the  figures  of  the  dividend  have 
been  annexed  to  the  remainder. 

If  any  partial  dividend  will  not  contain  the  divisor,  ivrite  a 
cipher  in  the  quotient,  then  annex  the  next  figure  of  the  divi- 
dend, and  proceed  as  before. 

If  there  is  a  remainder  after  the  last  division,  write  it  after 
the  quotient,  or  with  the  divisor  under  it  as  part  of  the  quotient. 

Pkoof.  —  Multiply  the  divisor  by  the  quotient,  and  to  the 
product  add  the  remainder,  if  any.  If  the  work  is  correct,  the 
result  will  equal  the  dividend. 

1 .  The  quotient  figure  may  generally  be  readily  obtained  by  finding 
how  many  times  the  first  figure  of  the  divisor  is  contained  in  the  first 
figures  of  the  dividend  tliat  will  contain  it,  making  allowance  for  the 
addition  to  the  product  from  the  second  figure  of  the  divisor. 

2.  If  the  product  of  the  divisor  by  the  quotient  figure  is  greater 
than  the  partial  dividend  from  which  it  is  to  be  subtracted,  the  quo- 
tient figure  is  too  large. 

3.  Each  remainder  must  be  less  than  the  divisor ;  otherwise  the 
quotient  figure  is  too  small. 

4.  When  there  is  no  remainder,  the  divisor  is  said  to  be  exact. 


128 


ELEMENTS   OF  ARITHMETIC. 


33. 

83)51875(625 
498 

207 
166 


34. 

74)42683(576^1 
370 

56S 
518 


35. 

52)45297(8715^ 
416 


369 
364 


415 
415 


503 
444 


59 
Find  the  quotients  of : 
36.   4202-11.  40.    2310 --42. 


37.  2556-12. 

38.  1386-21. 

39.  1488-31. 


41.  3796-52. 

42.  6840-15. 

43.  1728-12. 


57 

"5 

44.  2747-41 

45.  5084-62. 

46.  3367-91. 

47.  3024-84. 


Divide  each   of   the    following   numbers   by  21,  32,  43, 
54,  65 : 


48.  38425. 

49.  65484. 

50.  94653. 

51.  45677. 

52.  84732. 


53.  64136. 

54.  99874. 

55.  20562. 

56.  71117. 

57.  40505. 


58.  35794. 

59.  13570. 

60.  88992. 

61.  46823. 

62.  95895. 


63.  703821. 

64.  492532. 

65.  669046. 

66.  749638. 

67.  838327. 


Divide  each  of  the  following  numbers  by  37,  28,  19, 
46,  35 : 

68.  37905.  72.  29823.     76.  19198.  80.  469832. 

69.  35911.  73.  50847.     77.  34440.  81.  269724. 

70.  98634.  74.  72556.           78.  66984.  82.  123461. 

71.  48208.  75.  96054.     79.  85745.  83.  777738. 


Divide : 

84.  50854  by  94. 

85.  56394  by  78. 


86.  559108  by  202. 

87.  1547250  by  2134. 


DIVISION. 

129 

Drill 

Tahle. 

A. 

B. 

a 

D. 

E. 

F. 

Cr. 

1. 

4 

7 

42 

294 

882 

4410 

9076352 

2. 

5 

4 

36 

144 

576 

3456 

4487063 

3. 

6 

3 

24 

72 

360 

3240 

6470342 

4. 

7 

8 

40 

320 

960 

7680 

1598788 

5. 

5 

7 

5(3 

392 

1568 

15680 

7421077 

6. 

2 

8 

72 

576 

1152 

4608 

6480713 

7. 

3 

3 

27 

81 

729 

5103 

2835108 

8. 

8 

7 

28 

196 

1960 

9800 

5306792 

9. 

4 

8 

04 

512 

1536 

3072 

4350787 

10. 

3 

5 

40 

200 

1800 

5400 

7863444 

11. 

0 

G      ^ 

^     30 

180 

7200 

28800 

6095102 

12. 

7 

9 

27 

243 

1701 

8505 

7453745 

13. 

9 

8 

48 

384 

768 

2304 

6666006 

14. 

3 

6 

36 

216 

864 

3456 

8580437 

15. 

8 

8 

56 

448 

2240 

4480 

4654380 

16. 

3 

7 

63 

441 

3528 

7056 

7820194 

17. 

9 

7 

49 

343 

3773 

15092 

9003068 

18. 

4 

6 

42 

252 

1008 

5040 

3905746 

19. 

8 

9 

54 

486 

2430 

7290 

2849370 

20. 

4 

5 

40 

200 

1000 

9000 

6470181 

21. 

9 

9 

36 

324 

972 

8748 

5783900 

22. 

0 

8 

24 

192 

768 

3840 

4300125 

23. 

5 

9 

81 

729 

1458 

3916 

1758374 

24. 

3 

8 

72 

576 

2304 

9216 

9999999 

25. 

5 

6 

54 

324 

1296 

6480 

9876543 

EXERCISES  UPON  THE  TABLE. 


ORAL  EXERCISES 

. 

112.    1.    ^xi?=?                 2.    C  -.  B  = 

? 

3.    G-1 

WRITTEN   EXERCISES. 

4.    C+2)  +  ^=?                 9.  i)xC=? 

14.   E-^B=? 

5.    C+Z)+^+F  =  ?        10.  Cxl0=? 

15.  E^  C=? 

6.  F  -  i>  =  ?                       11.  2)  X  400  zz:  ? 

16.  F^  C=? 

1.  F-  E=?                       12.  D  X  203  =  ? 

17.   F^  D=? 

8.  DxB=?                       13.  D^B=? 

18.  100000  -  F 

19.    G  ^  D  =  ?                         21.   a  - 

{D  +  E+  F)=2 

20.  Fx  D  =  ?                        22.  Ax 

B 

X  C=  ?    . 

EL.    OF    AR.  —  9 

130  ELEMENTS  OF   ARITHMETIC. 

REVIEW  EXERCISES. 
ORAL    EXERCISES. 
113.  1.  A  bookseller  bought  15  books  for  $  30  and  sold  them 
at  a  gain  of  $  15.     How  much  apiece  did  he  get  for  them  ? 

2.  A  miller  sold  8  barrels  of  flour  at  $  5  a  barrel  and 
bought  with  the  proceeds  cloth  at  f  4  per  yard.  How  many 
yards  did  he  purchase  ? 

3.  The  cost  of  10  tons  of  coal  was  $  50,  but  I  sold  it  at 
a  gain  of  $  20.     How  much  did  I  receive  per  ton  ? 

4.  A  bookseller  bought  12  dictionaries^  at  $  8  apiece 
and  sold  them  at  $  10  apiece.     How  much  did  he  gain  ? 

5.  A  steam  yacht  ran  up  a  river  for  8  hours  at  10  miles 
per  hour,  and  it  came  down  in  5  hours.  What  was  its  rate 
per  hour  coming  down  ? 

6.  The  cost  of  5  tons  of  coal  at  f  5  per  ton  was  f  7  less 
than  the  cost  of  8  cords  of  wood.  What  did  the  wood 
cost  per  cord  ? 

7*  The  expense  of  sending  delegates  to  a  convention 
was  $  65.  If  the  entire  expense  had  been  $  1  less,  the  ex- 
pense to  each  member  would  have  been  $  2.  How  many 
members  were  there  ? 

8.  A  man  owed  a  debt  amounting  to  $  84.  If  he  paid 
$  60  of  the  sum  in  5  months,  in  how  long  a  time  will  he 
pay  it  all  at  that  rate  ? 

9.  A  farmer  sold  a  grocer  12  pounds  of  butter  at  20^ 
a  pound  and  took  in  exchange  brooms  at  24  cents  apiece. 
How  many  brooms  did  he  get  ? 

10.  Two  boys  set  out  from  the  same  place  and  traveled 
in  the  same  direction,  Arthur  traveling  5  miles  per  hour 
and  Bennie  3  miles  per  hour.  How  soon  will  Arthur  be  18 
miles  ahead  of  Bennie  ? 

11.  How  far  has  each  traveled  when  Arthur  is  24  miles 
ahead  ? 


DIVISION. 


12.  Suppose  they  travel  in  opposite  directions  at  those 
rates,  how  far  apart  will  they  be  in  8  hours  ? 

13.  In  how  many  days  can  4  men  do  as  much  work  as  8 
men  can  do  in  6  days  ? 

14.  Howard's  father  gave  him  an  acre  of  land  to  till  so 
that  he  might  earn  his  own  pocket  money.  He  sold  the 
products  of  the  land  for  $  38,  but  he  paid  $  5^  for  seeds  to 
plant  and  $  2i  for  help.     How  much  money  had  he  left  ? 

15.  A  boy  bought  6  lead-pencils  for  24j^  and  exchanged 
them  for  some  erasers  at  3p  each.  How  many  erasers  did 
he  get  ? 

16.  A  man  bought  a  carriage  for  $  120.  After  spending 
$  40  in  repairing  and  painting  it,  he  sold  it  for  $  150.  Did 
he  gain  or  lose,  and  how  much  ? 

17.  A  newsboy  bought  20  papers  for  30  cents  and  sold 
them  at  a  gain  of  30  cents.  For  how  much  apiece  did  he 
sell  them  ? 

18.  Henry  bought  6  car-tickets  for  25  cents.  How  many 
could  he  buy  for  50  cents  ? 

19.  A  clerk  saved  $  20  a  month.  In  how  many  months 
could  he  save  enough  to  buy  a  library  worth  $  240  ? 

20.  A  jeweler  bought  12  watches  for  $  72  and  sold  them 
at  a  gain  of  $  3  each.  At  what  price  did  he  sell  each  of 
them? 

21.  The  wages  of  a  machinist  were  $21  per  week,  and 
his  expenses  were  $  10.  In  how  many  weeks  can  he  pay  a 
debt  of  $  75  and  deposit  $  46  in  a  savings  bank  ? 

WRITTEN    EXERCISES. 

114.  1.  If  a  man  paid  ^  7564  for  124  horses,  how  much 
did  each  cost  ? 

2.  At  $  9  a  yard,  how  many  yards  of  velvet  can  I  buy 
for  $  207  ? 

3.  How  many  weeks  are  there  in  1736  days  ? 


132  ELEMENTS  OF  ARITHMETIC. 

4.  If  12  horses  cost  $  1560,  what  will  one  horse  cost? 
What  will  5  horses  cost  at  the  same  rate  ? 

5.  How  many  dozen  eggs  are  there  in  1728  eggs  ? 

6.  Divide  $  800.25  equally  among  15  men. 

7.  In  an  orchard  there  are  25  rows  of  trees.  If  there 
are  3200  trees  in  all,  how  many  trees  are  there  in  a  row  ? 

8.  How  many  times  26  bushels  are  there  in  22,568 
bushels  ? 

9.  How  many  acres  of  land  will  $22,575  buy,  at  $75 
an  acre  ? 

10.  In  64  loads  there  were  143,360  pounds.  How  many 
pounds  were  there  in  a  load  ? 

11.  How  many  days  are  there  in  3600  hours  ?  In  as 
many  minutes  ? 

12.  How  many  hours  are  there  in  4320  minutes  ?  How 
many  days  ? 

13.  What  number,  multiplied  by  23,  will  give  36,087? 

14.  Product,  62,397;  multiplier,  27;  find  the  multipli- 
cand. 

15.  If  the  product  of  two  factors  is  21,015,  and  one  fac- 
tor is  45,  what  is  the  other  factor  ? 

16.  If  you  weigh  1476  ounces,  how  many  pounds  (of  16 
ounces)  do  you  weigh  ?  How  many  pounds  does  your  sister 
weigh,  her  weight  being  133  ounces  less  than  yours  ? 

17.  If  the  receipts  of  a  railroad  for  one  year  (52  weeks) 
are  $  2,683,928,  how  much  are  they  per  week  ? 

18.  The  combined  wealth  of  405  men  is  $1,247,400. 
How  much  would  each  have  if  the  money  were  equally 
divided  among  them  ? 

19.  How  many  schooners,  each  carrying  8700  bushels  of 
wheat,  will  be  required  to  carry  843,900  bushels  ? 

20.  Suppose  that  two  cans  of  equal  size  together  hold 
376  pints.     How  many  gallons  are  there  in  each  ? 

21.  How  many  poor  families  may  be  supplied  from  37 


DIVISION".  133 

barrels  of  flour,  allowing  28  pounds  to  each,  a  barrel  of 
flour  weighing  196  pounds  ? 

22.  How  many  thousands  make  one  million  ? 

23.  If  a  railroad  company  pays  $  1750  for  constructing  a 
car,  how  many  cars  can  be  made  for  $  75,250  ? 

24.  The  average  daily  receipts  of  a  ferry-boat  are  $  325. 
In  how  many  days  will  the  receipts  amount  to  $  76,050  ? 

25.  The  stock  of  a  railroad  company  is  $>  2,565,000.  If 
it  is  divided  into  shares  of  $  500  each,  how  many  shares 
are  there  ? 

26.  Mount  Everest,  in  Asia,  is  said  to  be  29,100  feet 
high.  Since  there  are  5280  feet  in  a  mile,  how  many  miles 
high  is  it  ? 

27.  In  what  number  is  224  contained  28  times  ? 

28.  Mr.  Smith  earns  $28  while  Mr.  Allen  earns  $15. 
How  much  will  Mr.  Smith  earn  while  Mr.  Allen  earns 
$120? 

29.  How  many  times  will  a  cart-wheel,  15  feet  in  cir- 
cumference, revolve  in  going  a  mile,  or  5280  feet  ? 

30.  How  many  posts  and  how  many  rails  will  be  required 
for  a  fence  312  feet  long,  if  the  posts  are  set  12  feet  apart, 
and  the  fence  is  4  rails  high  ? 

31.  From  a  farm  containing  1000  acres,  there  were  sold 
at  one  time  274  acres ;  at  another  time,  238  acres.  How 
many  acres  remained  unsold  ? 

32.  Divide  $432  between  Mr.  A.  and  Mr.  B.,  so  that  ]Mr. 
B.  will  have  $  16  more  than  Mr.  A. 

33.  At  an  election,  the  sum  of  the  votes  received  by  two 
opposing  candidates  was  6742.  If  the  successful  candidate 
received  328  more  votes  than  his  opponent,  how  many  votes 
did  each  receive  ? 

34.  Into  a  tank  w^ere  poured  76  barrels  of  kerosene,  of 
43  gallons  each.  If  2489  gallons  are  drawn  from  the  tank, 
how  many  gallons  are  left? 


184  ELEMENTS   OF   ARITHMETIC, 

35.  If  I  buy  8  boxes  of  starch,  each  box  containing  24 
pounds,  at  8  cents  a  pound,  and  offer  in  payment  a  $  20  bill, 
what  change  shall  I  receive  ? 

36.  Elsie  and  Lottie  have  together  $5.70,  and  Lottie  has 
40  cents  more  than  Elsie.     How  much  money  has  each  ? 

37.  If  14  men  can  do  a  piece  of  work  in  56  days,  how 
long  will  it  take  2S  men  to  do  half  the  work  ? 

38.  I  bought  two  horses.  The  first  cost  $312,  and  the 
second  f  40  less  than  the  first.  How  much  did  the  two 
horses  cost  ? 

39.  If  19  tons  of  coal  run  an  engine  703  miles,  how  far 
will  17  tons  run  it  ? 

40.  A  farmer  exchanged  4  cows,  worth  $79  each,  for  a 
span  of  horses.     What  were  the  horses  worth  apiece  ? 

41.  A  horse  cost  $235,  a  carriage  $219,  and  a  hack  3 
times  as  much  as  both.     What  did  all  three  cost  ? 

42.  A  merchant  bought  42  bales  of  cloth,  each  bale  con- 
taining 36  pieces,  and  each  piece  38  yards,  at  9  cents  a  yard. 
He  sold  the  whole  at  11  cents  a  yard.  How  much  did  he 
gain  ? 

43.  If  I  take  764  bushels  from  2000  bushels  twice,  how 
much  will  remain  ? 

44.  A  drover  with  $  2144  bought  as  many  horses  as  pos- 
sible for  $  165  each,  and  spent  the  remainder  for  sheep  at 
$  4  a  head.     How  many  of  each  did  he  buy  ? 

45.  HoAv  many  times  945  will  make  23,625  ? 

46.  Twenty  men  built  a  school-house  in  56  days.  In  how 
many  days  could  70  men  have  built  it  ? 

47.  A  man  bought  163  barrels  of  flour  at  $9  a  barrel; 
15  barrels  were  spoiled,  and  the  remainder  he  sold  at  $  11  a 
barrel.     Did  he  gain  or  lose,  and  how  much  ? 

48.  A  school  uses  36  crayons  a  day.  Since  a  gross  is  12 
dozen,  how  many  gross  will  it  use  in  40  weeks,  provided 
there  is  no  school  on  Saturday  ? 


DIVISION.  135 

49.  If  it  takes  5  yards  of  cloth  to  make  a  pair  of  shirts, 
what  will  24  pairs  cost  at  15  cents  per  yard  for  the  cloth, 
45  cents  apiece  for  bosoms,  wristbands,  and  buttons,  and  75 
cents  apiece  for  making  ? 

50.  A  clerk  has  a  salary  of  f  42  a  month,  and  his  expenses 
are  f  27  a  month.  How  many  years  will  it  take  him  to  lay 
up  1 900  ? 

51.  Find  the  cost  of  267  yards  of  cotton  at  13  cents  a 
yard,  and  twice  as  much  at  12  cents  a  yard. 

52.  There  are  in  a  certain  school  37  pupils  14  years  old; 
76  pupils  12  years  old ;  114,  11  years  old  ;  149,  10  years 
old ;  and  168,  9  years  old.     What  is  their  average  age  ? 

53.  A  man  paid  $375  for  a  piano,  $13  for  freight  and 
cartage,  and  $2  for  tuning  it.  He  then  rented  it  for  7 
quarters  at  $  15  a  quarter,  and  afterwards  sold  it  for  $  325. 
Did  he  gain  or  lose,  and  how  much  ? 

54.  Two  men  leave  the  same  place  and  travel  in  opposite 
directions,  one  at  the  rate  of  30  miles  a  day,  and  the  other 
at  the  rate  of  37  miles  per  day.  How  far  will  they  be  apart 
at  the  end  of  14  days  ? 

55.  If  the  men  traveled  in  the  same  direction,  how  far 
apart  would  they  be  in  the  same  time  ? 

56.  A,  B,  and  C  have  together  $198.  A  and  C  have 
$  145.     A  and  B  have  $  99.     How  much  has  each  ? 

57.  Howard,  Eoy,  and  Jamie  went  fishing,  and  together 
caught  77  fish.  Of  these,  Howard  and  Jamie  together 
caught  58.  Howard's  and  Eoy's,  when  put  together,  num- 
bered 54.     How  many  did  each  catch  ? 

58.  A  farmer  sold  18  calves  at  the  rate  of  3  for  $33 
How  much  did  he  get  for  them  ? 

59.  If  117  tons  of  coal  are  worth  $702,  how  much  are 
235  tons  worth  ? 

60.  If  12  barrels  of  flour  are  worth  $132,  what  will  36 
barrels  cost  ? 


136  ELEMENTS   OF   ARITHMETIC. 

61.  If  a  farmer  receives  f  2275  for  13  horses,  for  how 
much  would  27  horses  sell  at  the  same  rate  ? 

62.  A's  farm  contains  227  acres;  B's  lacks  12  acres  of 
being  twice  as  much ;  C's  lacks  42  acres  of  being  3  times  as 
much  as  B's ;  and  D's  contains  as  much  as  A's  and  C's. 
How  many  acres  are  there  in  each  farm  ? 

63.  A  farmer  wished  to  obtain  $  260.  He  sold  72  bushels 
of  wheat  at  $1.50  per  bushel,  and  enough  apples  at  $2  per 
barrel  to  obtain  the  sum  required.  How  many  barrels  of 
apples  did  he  sell  ? 

64.  If  a  bookkeeper  receives  $1400  per  year  for  his  ser- 
vices, and  his  expenses  are  $  840,  in  what  time  can  he  save 
enough  to  buy  32  acres  of  land  at  $  140  per  acre  ? 

65.  A  man  starting  out  on  a  journey  took  $200.  He 
paid  for  railroad  fare  $  67 ;  for  berth  in  sleeping  car,  4 
days,  $  2  a  day  ;  for  hotel  bills,  15  days,  $  3  a  day  ;  and  for 
other  expenses,  $  25.     How  much  money  had  he  left  ? 

66.  The  greater  of  two  numbers  is  4056,  and  their  differ 
ence  is  3650.     What  is  the  less  number  ? 

67.  The  divisor  is  534,  and  the  quotient  43.  What  u 
the  dividend  ? 

68.  The  subtrahend  is  34,203,  and  the  remainder  is  8706 
What  is  the  minuend  ? 

115.  The  parenthesis,  (  ),  shows  that  the  numbers  in- 
cluded within  it  are  to  be  subjected  to  the  same  operation. 

Thus,  (15  —  3)  X  3  means  tliat  15  —  3,  or  12,  is  to  be  multiplied  by 
3 ;  but  15  —  3  X  3  means  that  15  is  to  be  diminished  by  3  x  3,  or  9. 

(7  +  2)  X  5  means  that  the  sum  of  7  and  2  is  to  be  multii)liecl  by 

5  ;  but  7  +  2x5  means  that  7  is  to  be  increased  by  2  x  5. 

7  +  5 
(7  +  5)-^  (j,   or    — -";—  ,    means  that  the  sum  of  7  aiul  5  is  to  be 
G 
divided  by  0. 

[(9  +  3)  x  2  -  3]  -^  7  means  that  the  sum  of  f)  and  3  is  to  be  multi- 
plied by  2,  the  product  diminished  by  3,  and  the  remainder  divided 
by  7. 


DIVISION.  137 

(7  X  8) -(5  X  5)  means  that  the  product  of  7  and  8  is  to  be 
diminished  by  the  product  of  5  and  5. 

116.  The  Vinculum, ,  or  brackets  [  ],  may  be  used 

instead  of  the  parenthesis. 

Thus,  5  —  3  X  3  or  [5  — 3J  x3  may  be  used  instead  of  (5  —  3)  x  3. 

117.  In  finding  the  value  of  expressions  in  which  the 
parenthesis,  vinculum,  or  brackets  are  used, 

First,  perform  the  operations  on  the  numbers  that  are  ivrit- 
ten  icithin parenthesis  and  brackets,  or  under  the  vinculum,  as 
indicated  by  the  signs.  Next,  multipily  and  divide,  as  indicated 
by  the  signs  x  and  -t-.  Finally,  add  and  subtract,  as  indi- 
cated by  the  signs  +  and  —. 

rind  the  values  of  the  following  : 

1.    (4  +  7)x5.  8.    l±^  +  ^-^ 


3  4 

2.  4  +  7x5.  9.    2x12-2x3. 

3.  (8-3)x2.  10.    2x12 -(2x3). 

4.  8-3x2.  11.    i8-(3x4)-(2x3)^ 

o 

5.  4  +  16-4.  12.    [(5  +  6)x4-5  x.S]x9. 

6.  (4 +  16) -4.  13.    (3  X  4)  X  5 -(9 +  7) -8. 
7o    (3  +  4)  X  (8-5).     14.    2  +  12-4+(8  +  8-4)-2„ 

15.  (312  +  36)— (381 -215) -65. 

16.  (214 -81) -(115 -18 +  6) +  10. 

17.  (413  _  200)  -  (118  -  24  -  4  +  6)  +  3. 

18.  (171  -  S(\)  -  (3^r4  +  27)  +  10. 

19.  (3x4)x0-(4  +  12)-f-4  +  5. 

20.  (5 +  2 +  6)  X  4 -(7 +  8 -3) -6 +  10. 

21.  "(36-7)x  5 +  (102 +  6) -9. 


22.    (99  -  3)  -  8  -  (86  +  10)  -  12  +  (3  +  6)  -  3. 


FACTORING. 


118.  1 .  What  is  the  product  when  3  and  2  are  multiplied 
together  ?     What  are  3  and  2  of  their  product  ?     (Art.  86.) 

2.  What  factors  will  produce  9  ? 

3.  What  numbers  when  multiplied  together  will  produce 
12  ?     "What  are  3  and  4,  or  6  and  2  of  their  product  ? 

4.  What  are  the  factors  of  20  ?     Of  36  ?     Of  15  ? 

5.  What  are  the  factors  of  27  ?     Of  25  ?     Of  32  ? 

6.  What  are  the  factors  of  21  ?     Of  33  ?     Of  50  ? 

7.  What  are  the  factors  of  63  ?  What  else  may  7  be 
called  of  63  besides  a  factor  ? 

8.  If  9  is  one  of  two  factors  of  18,  what  is  the  other 
factor  ?  If  3  is  one  of  the  factors  ?  If  6  is  one  of  the 
factors  ? 

9.  What  numbers  will  exactly  divide  18  ?     25?     36  ? 

10.  Give  the  exact  divisors  of  42;  96;  35;  50;  27;  72. 

11.  Give  the  factors  of  36 ;  40;  48;  70;  80. 

12.  Give  the  exact  divisors  of  44 ;  56;  64;  84;  96. 

13.  Name  the  exact  divisors  of  49;  88;  63;  24;  27. 

14.  What  numbers  between  0  and  10  cannot  be  exactly 
divided  by  any  number  except  themselves  and  1  ?  What 
numbers  between  10  and  20  ?     Between  20  and  30  ? 

15.  What  numbers  between  0  and  10  can  be  exactly 
divided  by  other  numbers  besides  themselves  and  1  ?  Be- 
tween 10  and  20  ?     Between  20  and  30? 

16.  Select  from  the  following  the  numbers  that  have  no 
exact  divisors  except  themselves  and  1:  35,  42,  63,  56,  61, 
47,  49,  81,  37,  26, 18,  45. 

138 


FACTORING.  139 

17.  Select  from  the  following  the  numbers  that  have 
exact  divisors  besides  themselves  and  1 :  24,  36,  41,  39,  27, 
45,  33,  37,  50,  44,  60,  71,  72. 

119.  A  number  that  expresses  whole  units  is  called  an 
Integer. 

Thus,  5,  27,  35  are  integers,  or  integral  numbers. 

120.  The  integers  which,  upon  being  multiplied  together 
will  produce  the  number,  are  called  Factors  of  the  number. 

Thus,  5  and  3  are  the  factors  of  15. 

121.  An  integer  which  will  divide  a  number  without  a 
remainder  is  called  an  Exact  Divisor  of  the  number. 

Thus,  2,  3,  6,  and  9  are  exact  divisors  of  18.  They  are  also  factors 
of  18. 

122.  A  number  that  has  no  exact  divisors  except  itself 
and  1  is  called  a  Prime  Number. 

Thus,  1,  3,  5,  7  are  prime  numbers. 

123.  A  number  that  has  exact  divisors  besides  itself  and 
1  is  called  a  Composite  Number. 

Thus,  24,  36,  40,  100  are  composite  numbers. 

124.  Factors  that  are  prime  numbers  are  Prime  Factors. 

Thus,  7  and  5  are  the  prime  factors  of  35. 

125.  A  number  that  is  exactly  divisible  by  2  is  called  an 
Even  Number. 

Thus,  8,  12,  20,  24  are  even  numbers. 

126.  A  number  that  is  not  exactly  divisible  by  2  is  called 
an  Odd  Number. 

Thus,  15,  21,  35,  43  are  odd  numbers. 

127.  The  process  of  separating  a  number  into  its  factors 
is  called  Factoring. 


140  ELEMENTS   OF   ARITHMETia 

TESTS  OP  DIVISIBILITY^ 

128.  1.  Make  a  list  of  numbers  from  1  to  126,  which 
have  2  for  one  or  more  of  their  factors  or  divisors,  iSTotice 
what  the  right-hand,  or  units'  figure,  of  each  is. 

2.  Make  a  list  of  numbers  from  1  to  100  which  have  5 
foi  one  or  more  of  their  factors  or  divisors.  Notice  what 
the  units'  figure  of  each  is. 

3.  Make  a  list  of  numbers  from  1  to  100  which  have  3 
for  one  or  more  of  their  factors  or  divisors.  Divide  the 
sum  of  the  digits  of  each  of  these  numbers  by  3,  and  notice 
the  remainder,  if  any. 

4.  Make  a  list  of  numbers  from  1  to  200  which  have  9 
for  one  or  more  of  their  factors  or  divisors.  Divide  the  sum 
of  the  digits  of  each  by  9,  and  notice  the  remainder,  if  any. 

129.  It  is  apparent,  therefore,  that: 
A  immher  is  divisible  by 

2  if  the  units'  fgure  is  2,  4,  6,  8,  or  0. 
5  if  the  units'  figure  is  0  or  5. 

3  if  the  sum  of  its  digits  is  divisible  by  3. 
9  if  the  sum  of  its  digits  is  divisible  by  9. 

130.  Tell  by  inspection  which  of  the  numbers,  on  page 
129,  columns  O  and  D,  are  divisible  by  2;  by  5 ;  by  3 ;  by  9. 

Which  in  column  E  are  divisible  by  2 ;  by  3 ;  by  5 ;  by  9. 

WRITTEN    EXERCISES. 

131.  1.    What  are  the  prime  factors  of  336? 

Explanation.—  Since  every  fac- 
tor of  a  number  is  a  divisor  of  it, 
we  may  find  the  prime  factors  of 
336  by  dividing  by  tlie  exact  divisors 
that  are  prime  numbers.  Dividing 
by  2,  we  find  the  factors  of  330  to 
be  2  and  168.  But  168  has  a  factor 
2,  and,  since  a  factor  of  a  factor  of 
a  number  is  a  factor  of  the  numbei 
itself,  we  continue  tlie  process,  and  obtain  tlie  prime  factors  of  336. 


2 

336 

2 

168 

2 

84 

2 

42 

3 

21 

7 

336  =  2  X  2  > 

<2xl 

?  X  3  X  7 

FACTORING. 

] 

Vha 

.t  are  the  prime 

!  factors 

2. 

Of  168? 

12. 

Of  625? 

22. 

Of  1280  ? 

3. 

Of  224  ? 

13. 

Of  912  ? 

23. 

Of  2625? 

4. 

Of  144  ? 

14. 

Of  832  ? 

24. 

Of  3204? 

5. 

Of  198  ? 

15. 

Of  990? 

25. 

Of  1155? 

6. 

Of  216  ? 

16. 

Of  1008? 

26. 

Of  2500  ? 

7. 

Of  484  ? 

17. 

Of  435  ? 

27. 

Of  1485? 

8. 

Of  576  ? 

18. 

Of  222? 

28. 

Of  1296? 

9. 

Of  432  ? 

19. 

Of  672? 

29. 

Of  2016  ? 

10. 

Of  672  ? 

20. 

Of  775  ? 

30. 

Of  2875? 

lie 

Of  396  ? 

21. 

Of  289  ? 

31. 

Of  2890? 

141 


CANCELLATION. 
ORAL   EXERCISES. 

132.  1.  How  many  times  is  2  times  5  contained  in  4 
times  5  ?  2  times  3  in  4  times  3  ?  2  times  any  number  in 
4  times  that  number  ? 

2.  How  many  times  is  4  times  7  contained  in  8  times  7  ? 

4  times  35  in  8  times  35  ?     4  times  a  certain  number  in  8 
times  the  same  number  ? 

3.  How   many   times  is   6x12   contained  in   18x12? 

5  X  23  in  15  X  23  ?     7  x  47  in  21  x  47  ? 

4.  What  is  the  quotient  of  (24  x  17)-^  (12  x  17)?  Of 
(63x24)-(9x24)?  Of  (48  x  61)- (24  x  61)?  Of  (36 
times  19)  divided  by  (18  times  19)  ? 

5.  In  determining  the  quotient,  what  numbers  may  be 
omitted  from  both  dividend  and  divisor  ? 

133.  From  the  solution  of  the  examples  given,  it  is 
evident  that : 

Rejecting  equal  factors  from  both  dividend  and  divisor  does 
not  alter  the  quotient. 


2 

2 

2 

2 

^ 

X0 

xgx 

w_ 

16 
''  3 

-  .^1 

% 
1 

x^ 
1 

X^  X 

1 

3 

-  ^3 

142  ELEMENTS   OF   ARITHMETIC. 


WRITTEN    EXERCISES. 

134.   1.   Divide  4x6x8xl0by2x3x4xl5. 

Explanation.  —  The  dividend  is 
written  above  the  divisor,  with  a  line 
between  them  as  in  division. 

Since  the  factors  2,  3,  and  4  are 
factors  of  4,  6,  and  8  in  the  dividend, 
we  reject  them  from  both  dividend  and  divisor,  leaving  the  factors  2, 
2,  and  2  in  the  dividend.  Since  the  factor  5  is  common  to  10  in  the 
dividend  and  to  15  in  the  divisor,  it  is  also  rejected,  leaving  2  in  the 
dividend,  and  3  in  the  divisor. 

The  product  of  the  uncanceled  factors  of  the  dividend  is  16,  and 
of  the  divisor  3.     Hence,  the  quotient  is  ^^,  or  16  -;-  3,  or  5^. 

2.    Divide  7  x  4  X  11  X  8  by  22  x  4  x  4  x  21. 

-I  Explanation,  —  In  this   example,   the 

.         .        ^        ^  common  factor  7  is  rejected  from  7  and  21, 

-*■        -^  ^  leaving   1    in  the   dividend   and  3  in  the 

T  X  ^  X  11  X  g  ^  x^       divisor. 

^^  X  ^  X  ^  X  ?J       ^  Next,  the  common  factor  4  is  removed 

<^       \       \       2t  ixom.  4  in  the  dividend  and  4  in  the  divisor. 

?  Then  the  common  factor  4  is  removed  from 

8  in  the  dividend  and  4  in  the  divisor,  leav- 
ing 2  in  the  dividend  and  1  in  the  divisor.    Then  the  common  factor 
11  is  rejected  from  11  in  the  dividend  and  from  22  in  the  divisor, 
leaving  1  in  the   dividend  and  2  in  the  divisor.    Then  the  common 
factor  2  is  removed  from  the  dividend  and  the  divisor. 
The  product  of  1  x  1  x  1  x  1  in  the  dividend  is  1. 
The  product  of  1  x  1  x  :l  x  3  in  the  divisor  is  3. 
Hence,  the  quotient  is  1  -^  3,  or  J. 

1.  When  the  factor  is  contained  once  in  any  number,  the  1  may 
be  omitted ;  but  it  is  better  to  write  it  until  the  process  is  clearly 
understood. 

2.  When  all  the  factors  of  both  divisor  and  dividend  are  canceled, 
the  quotient  is  1,  for  the  dividend  will  then  exactly  contain  the 
divisor  once. 

Finxl  the  value  of : 

4x5x7x9  ,       8x9x12x16 


3. 


2x2x6x7x3  *    4x3x5x6  X  20 


FACTORING.  143 

2x3x8x12x24  ^     15  X  20  X  25  X  27 

6  X  4  X  36  X  4     *  '    10  X  15  X  18  X  25* 

18  X  24  X  32  X  36  g     25  x  30  x  40  x  35 

*      9  X  48  X  4  X  18  '  '    20  X  15  X  18  X  7  * 

Divide,  using  cancellation ; 

9.  40  X  48  X  54  X  60  by  30  x  24  x  72  x  3. 

10.  30  X  49  X  64  X  25  by  15  x  35  x  24  x  10. 

11.  12  X  60  X  36  X  70  by  28  X  48  X  6  X  5. 

12.  32  X  36  X  33  X  45  by  24  x  30  x  44  x  9. 

13.  30  X  36  X  24  x  42  by  45  x  27  x  8  x  28. 

14.  27  x  32  X  45  x  36  by  18  X  24  X  9  X  6. 

15.  45  X  28  X  36  X  56  by  49  X  4  X  18  X  18. 

16.  27  X  28  X  35  X  49  by  18  x  21  x  40  x  28. 

17.  45  X  60  x  63  x  28  by  27  x  20  x  25  x  21. 

18.  55  X  36  X  27  x  42  by  12  x  25  x  35  x  33. 

19.  36  X  64  X  25  x  40  by  32  x  50  x  18  x  10. 

20.  5()  X  18  X  32  X  49  by  16  x  36  x  42  x  28. 

21.  32  X  45  X  72  X  15  by  64  x  18  x  36  x  30. 

22.  64  X  15  X  63  X  24  by  30  x  21  x  56  x  12. 

23.  44  X  20  X  50  x  36  by  55  x  10  x  25  x  18. 

24.  40  X  48  X  21  X  18  by  56  x  36  x  35  x  24. 

25.  40  X  27  X  32  X    3  by  5  x  8  x  12  x  8  x  27. 

26.  A  lady  bougiit  9  yards  of  ribbon  at  56  cents  per 
yard,  but  exchanged  it  for  other  ribbon  at  32  cents  per 
yard.     How  many  yards  did  she  get  ? 

27.  At  the  rate  of  $129  for  27  acres  of  land,  what  will 
180  acres  cost  ? 

28.  At  what  price  per  yard  will  5  bales  of  cloth,  contain- 
ing 12  pieces  of  42  yards  each,  pay  for  50  rolls  of  carpeting, 
of  75  yards  each,  at  $  2.10  per  yard  ? 

29.  A  merchant  purchased  13  firkins  of  butter,  each  con- 
taining 39  pounds,  at  32  cents  a  pound,  and  paid  for  it  with 


144  ELEMENTS  OF   ARITHMETIC. 

4  patterns  of  silk  of  13  yards  each.  How  mucli  was  tlie 
silk  worth  per  yard  ? 

30.  A  grocer  exchanged  27  chests  of  tea,  each  containing 
48  pounds,  worth  40  cents  a  pound,  for  18  pieces  of  dress 
goods,  each  containing  60  yards.  What  was  the  price  per 
yard  of  the  dress  goods  ? 

31  o  A  stationer  had  28  packages  of  cards,  each  contain- 
ing 24  cards,  which  he  exchanged  for  21  packages  of  note 
paper,  each  containing  24  sheets,  worth  2  cents  per  sheet. 
AVhat  were  the  cards  worth  apiece  ? 

32.  Four  farms,  each  containing  80  acres,  worth  $  72 
per  acre,  were  exchanged  for  5  farms,  each  containing  96 
acres.  What  was  the  value  per  acre  of  the  farms  received 
in  exchange  ? 

33.  A  farmer  exchanged  8  bushels  of  wheat,  at  f  2  per 
bushel,  for  3  pieces  of  muslin,  worth  10  cents  per  yard. 
How  many  yards  were  there  in  each  piece  ? 

34     5x8x7x12x16x24x10x9^^ 
3x16x18x4x20x6x5x8 

_     9  X  8  X  11  X  16  X  15  X  35  X  42  X  18  _  ^ 


36. 


37. 


38. 


39- 


40. 


6x21x7x5x25x32x4x9 

12  X  9  X  30  X  25  X  16  X  36  X  48 
18  X  6  X  15  X  35  X  72  X  24  X  9 


_  ? 


=  9 


42  X  35  X  18  X  24  X  60  X  64  X  72 
36  X  32  X  12  X  25  X  30  X  21  X  5 

44  X  20  X  36  X  54  X  65  X  27  X  48 

24  X  40  X  27  X  72  X  13  X  45  X  72 

45  X  60  X  63  X  28  X  35  X  24  X  50^  ^ 

25  X  21  X  56  X  21  X  40  X  48  X  36 

36  X  48  X  21  X  35  X  42  X  96  X  56  ^  y 
28  X  84  X  42  X  24  X  70  X  14  X  9 


FRACTIONS. 


135.    1.  Into  Llow  many  parts  are  these  squares  divided? 

2.  How  do  the  parts  compare  as  to  size  ? 

3.  What  is  each  part  called  ? 

4.  What  is  a  half  of  anything  ? 

5.  How  many  halves  are  equal  to  a  whole  one  ? 

6.  How  many  halves  are  there  in  2  apples  ?  In  3  pears  ? 
In  5  peaches  ?  In  1  orange  and  a  half  ?  In  2  lemons  and 
a  half  ?     In  4  and  a  half  plums  ? 

7.  Two  half  melons  are  equal  to  how  many  whole 
melons  ?  4  half  melons  ?  6  half  melons  ?  3  half  melons 
are  equal  to  how  many  whole  melons^  and  how  many  half 
melons  besides  ?     5  half  melons  ? 

8.  Draw  a  line.  Divide  it  into  halves.  Then  divide 
each  half  into  two  equal  parts. 

9.  Into  how  many  parts  is  the  whole  line  divided? 

10.  What  is  each  part  called  ? 

11.  What  do  you  call  2  of  the  parts  ?  3  of  the  parts? 
How  many  thirds  are  equal  to  a  whole  thing? 

12.  How  many  fourths  are  there  in  any  whole  thing? 

13.  How  many  fourths  are  there  in  2  circles  ?  In  3 
squares  ?  In  1  line  and  1  fourth  of  a  line  ?  In  2  lines  and 
3  fourths  of  a  line  ? 

EL.    OF  AR.  — 10  145 


146  ELEMENTS   OF  ARITHMETICo 

14.  How  many  fourths  of  an  apple  are  there  in  half  of  an 
apple  ? 

15.  What  part  of  a  half  is  a  fourth  ? 

16.  Which  is  the  greater^  1  half  of  a  yard  or  2  fourths  of 
a  yard  ?     1  pineapple  or  3  fourths  of  a  pineapple  ? 

17.  How  many  whole  apples  are  equal  to  8  fourths  of 
an  apple  ?  To  12  fourths  ?  How  many  whole  apples  and 
how  many  fourths  besides  are  equal  to  5  fourths  ?  To 
7  fourths  ?     To  9  fourths  ?     To  10  fourths  ? 

18.  Draw  a  circle.  Divide  it  into  fourths.  Then  divide 
each  fourth  into  2  equal  parts. 

19.  Into  how  many  equal  parts  is  the  circle  divided  ? 

20.  What  is  each  part  called? 

21.  What  do  you  call  2  of  the  parts?  3  of  the  parts f 
5  of  the  parts  ?     6  of  the  parts  ?     7  of  the  parts  ? 

22.  How  many  eighths  are  there  in  a  pie  ?  In  2  pies  ? 
In  2  pies  and  1  eighth  of  a  pie  ?  In  3  pies  and  3  eighths  of 
a  pie  ?  In  1  fourth  of  a  pie  ?  In  3  fourths  of  a  pie  ?  In 
1  pie  and  3  fourths  of  a  pie  ?  In  half  a  pie  ?  In  2  and  a 
half  pies  ? 

23.  How  many  whole  sticks  of  candy  are  equal  to  8 
eighths  ?  16  eighths  ?  How  many  whole  sticks  of  candy 
and  how  many  eighths  besides  are  equal  to  10  eighths  ? 
To  15  eighths  ?    To  17  eighths  ?     To  20  eighths  ? 

^  24.    What  is  each  part  of  this   line 

called  ? 
25  o    What  do  you  call  2  of  these  parts  ? 

26.  How  many  thirds  are  there  in  the  whole  of  anything  ? 

27.  How  many  thirds  are  there  in  3  lines  ?  In  5  lines  ? 
In  2  lines  and  1  third  of  a  line  ?     In  4  and  2  thirds  lines  ? 

28.  If  I  have  6  thirds  of  orange,  to  how  many  whole 
oranges  is  that  equal  ?     9  thirds  ?     12  thirds  ? 

29.  How  is  this  line  divided  ? 

30.  What  is  each  part  called  ?   What 


FRACTIONS.  147 

do  you  call  2  parts  ?     What  other  name  will  express  the 
same  part  of  the  line  that  2  sixths  expresses  ? 

31.  What  do  you  call  3  parts  of  this  line?  What  other 
name  will  express  the  part  of  the  line  that  3  sixths 
expresses  ?  , 

32.  What  do  you  call  4  parts  of  this  line  ?  What  other 
name  will  express  the  part  of  the  line  that  4  sixths 
expresses  ? 

33.  How  many  sixths  are  there  in  2  sheets  of  paper? 
3  sheets  ?     4  sheets  ?     5  sheets  ? 

34.  Which  is  the  greater,  1  third  of  a  pie,  or  1  sixth  of 
a  pie? 

35.  What  part  of  a  third  is  a  sixth  ? 

36.  How  many  sixths  are  there  in  two  thirds?  In  1 
peach  and  1  third  of  a  peach  ?  In  2  and  a  half  peaches  ? 
In  3  and  5  sixths  peaches  ? 

37.  How  many  units  and  how  many  sixths  besides  are 
there  in  7  sixths  ?     9  sixths  ?     12  sixths  ?     14  sixths  ? 

38.  Draw  a  line.  Divide  it  into  sixths.  Now  divide 
each  sixth  into  2  equal  parts. 

39.  Into  how  many  equal  parts  is  the  line  divided  ? 

40.  What  is  each  part  called  ? 

41.  How  many  twelfths  are  there  in  the  whole  of  any- 
thing ? 

42.  How  many  twelfths  of  a  line  are  equal  to  a  sixth? 
To  a  third  ?  To  a  half  ?  To  2  thirds  ?  To  5  sixths  ?  To  1 
and  5  sixths  ?     To  2  and  a  half  ?    To  2  and  2  thirds  ? 

43.  How  many  units  and  how  many  twelfths  besides  are 
equal  to  15  twelfths  ?     To  18  twelfths  ?     To  20  twelfths  ? 

44.  What  part  of  a  sixth  is  a  twelfth  ?  What  other 
name  will  express  the  part  of  the  line  that  2  twelfths 
expresses?  4  twelfths?  6  twelfths?  8  twelfths?  10 
twelfths  ? 

45.  Draw  three  parallel  lines  of  just  the  same  length. 


148  ELEMENTS   OF   ARITHMETIC, 

46o    Divide  the  first  one  into  halves,  the  second  one  into 
fourths,  and  the  third  one  into  eighths. 

47.  Draw   three   other   parallel  lines  of  just  the  same 
length  as  the  first  group. 

48.  Divide  the   first   one    into   thirds,  the   second   into 
sixths,  and  the  third  into  twelfths. 

49.  Which  is  the  greater,  1  half  or  1  third  ? 

50.  Which  is  the  greater,  1  half  or  2  thirds  ? 

51.  When  you  divide  the  halves  into  2  equal  parts,  what 
parts  of  the  whole  do  you  have  ? 

52.  When  you  divide  the  thirds  into  2  equal  parts,  what 
parts  of  the  whole  do  you  have  ? 

53.  One  third  is  equal  to  how  many  sixths  ? 

54.  Two  thirds  are  equal  to  how  many  sixths  ? 

55.  Three  thirds  are  equal  to  how  many  sixths? 

56.  What  is  1  half  of  1  half  ?     1  half  of  1  third  ? 

57.  Which  is  the  greater,  1  third  or  1  fourth  ? 

58.  Which  is  the  greater,  1  fourth  or  1  sixth  ? 

59.  Which  is  the  greater,  1  third  or  2  sixths  ? 

60.  What  parts  of  the  whole  do  you  get  by  dividing  the 
fourths  into  2  equal  parts  ? 

61.  One  fourth  is  equal  to  how  many  eighths  ? 

62.  Two  fourths  are  equal  to  how  many  eighths  ? 

63.  Three  fourths  are  equal  to  how  many  eighths  ? 

64.  Which  is  the  greater,  1  sixth  or  1  eighth  ? 

65.  How  many  sixths  are  equal  to  a  half? 

66.  How  many  eighths  are  equal  to  a  half  ? 

67.  Which  is  the  greater,  3  sixths  or  3  eighths  ? 

68.  What  parts  of  the  whole  do  you  get  by  dividing  the 
sixths  into  2  equal  parts  ? 

69.  One  sixth  is  equal  to  how  many  twelfths  ? 

70.  Two  sixths  are  equal  to  how  many  twelfths  ? 

71.  Three  sixths  aie  equal  to  liow  many  twelfths?    Four 
sixths  ?     5  sixths  ?     6  sixths  ? 


FRACTIONS.  149 

72.  Which  is  the  greater,  1  twelfth  or  1  eighth  ? 

73.  How  many  twelfths  are  equal  to  a  half  ? 

74.  How  many  eighths  are  equal  to  a  half  ? 

75.  Which  is  the  greater,  6  twelfths  or  6  eighths  ? 

76.  When  anything  is  divided  into  o  equal  parts,  what  is 
each  part  called  '■ 

77.  What  do  you  call  3  of  the  parts  ? 

78.  How  many  fifths  are  there  in  the  whole  of  anything  ? 

79.  Draw  a  line  and  divide  it  into  fifths.  Divide  each 
fifth  into  2  equal  parts.  Into  how  many  equal  jparts  is  the 
line  divided  ? 

80.  What  is  each  part  called? 

81.  How  many  tenths  are  there  in  the  whole  of  anything  ? 

82.  How  many  tenths  are  equal  to  a  fifth  ? 

83.  Draw  a  line  and  divide  it  into  thirds.  Divide  each 
third  into  3  equal  parts.  Into  how  many  equal  parts  have 
you  divided  the  line  ?     What  is  each  part  called  ? 

84.  How  many  ninths  are  equal  to  a  third  of  the  line  ? 

136.  One  or  more  of  the  equal  parts  of  anything  is  a 
Fraction. 

137.  Two  numbers  written  one  above  the  other  with  a 
line  between  them  are  used  to  express  a  fraction. 

138.  The  number  which  shows  into  how  many  equal 
parts  the  unit  has  been  divided  is  called  the  Denominator. 

It  is  written  below  the  line. 

Thus,  ill  the  fraction  f,  5  is  the  denominator.  It  shows  that  the 
unit  has  been  divided  into  five  equal  parts. 

139.  The  number  which  shows  how  many  parts  form  the 
fraction  is  called  the  Numerator. 

It  is  written  above  the  line. 

Thus,  in  the  fraction  f ,  3  is  the  numerator.  It  shows  that  the  frac- 
tion contains  3  of  tlie  5  equal  parts  into  which  tlie  unit  was  divided. 


150  ELEMENTS   OF  ARITHMETIC. 

140.  The    numerator   and   denominator   are   called  tke 
Terms  of  a  Fraction. 

141.  A  fraction  in  which  the  numerator  is  less  than  the 

denominator  is  called  a  Proper  Fraction. 

Thus,  I,  I,  If  are  proper  fractions. 

The  value  of  a  proper  fraction  is,  tlierefore,  less  than  1. 

142.  A  fraction  in  which  the  numerator  equals  or  exceeds 

tlie  denominator  is  called  an  Improper  Fraction. 

Thus,  f,  I,  If  are  improper  fractions. 

The  value  of  an  improper  fraction  is,  therefore,  1  or  more  than  1. 

143.  A  number  expressed  by  an  integer  and  a  fraction  is 
a  Mixed  Number. 

Thus,  6|,  8^,  6\}  are  mixed  numbers. 

144.  A  fraction  may  be  regarded  as  expressing  unexecuted 
division. 

Thus,  ^  is  equal  to  15  ^  4  ;  ^A  is  equal  to  24  ■-  6. 

145.  1.    Eead  and  interpret  the  expression  ■^^. 
Explanation. — y\  represents  5  of  the  13  equal  parts  into  which 

the  unit  has  been  divided.     It  also  represents  one  thirteenth  of  6,  and 
5  divided  by  13.     It  is  readj'??;e  thirteenths. 

Eead  and  interpret  the  following  expressions  : 

2.  f.  5.    f  8.    ^.  11.    H-  14.  If. 

3.  |.  6.    f.  9.    ^.  12.    if  15.  li. 

4.  1.  7.    f.  10.    A.  13.    H-  16.  If 

Express  by  figures: 

17.  Five  sevenths.     Ten  fourteenths.     Seven  twelfths, 

18.  Eight  ninths.     Nine  tenths.     Five  thirteenths. 

19.  Nineteen  twenty-fifths.     Thirty-eight  fortieths, 

20.  Thirty  thirty-tiinths.     Forty-one  sixty-eighths. 

21.  Forty-six  eightieths.     Twenty-seven  forty-fifths. 

22.  Five  two-hundred-sixths.  Seven  three-hundred-ninths, 

23.  One  eighth  of  17.     One  thirty-fifth  of  59. 

24.  Fifteen  divided  by  1.7.     Twenty-four  divided  by  83. 


FRACTIONS.  151 

REDUCTION  OF  FRACTIONS. 

146.  To  reduce  integers  or  mixed  numbers  to  fractions, 

1.  A  man  received  $  2  for  a  day's  work.    If  he  was  paid 
in  half  dollars,  how  many  did  he  get  ? 

2.  How  many  quarter  dollars  are  equal  to  $  2  ? 

3.  How  many  sixths  are  there  in  2  pies  ?     Eiglitlis  ? 

4.  In  three  things  how  many  halves  are  there  ?   Fourths  ? 
Eighths?    Thirds?    Sixths?    Twelfths?   Fifths?   Tenths? 

5.  How  many  fourths  are  there  in  2  ?     In  2J-  ?     In  2|? 

6.  How  many  eighths  are  there  in  1  ?     In  3f  ?     In  3|-  ? 

7.  How  many  sixths  are  there  in  3  ?     In  If  ?     In  2|? 

8.  How  many  fifths  are  there  in  2  ?     In  2f  ?     In  34  ? 

9.  How  many  sevenths  are  there  in  1?    In  1^?    In  4f  ?. 

147.  The  process  of  changing  the  form  of  fractions  with- 
out changing  their  value  is  called  Reduction  of  Fractions. 

WRITTEN   EXERCISES, 

148.  1.    Eeduce  12f  to  eighths. 

-j9  _'96_  Explanation.  — Since  in  1  there  are  8  eighths,  in 

9  6  ~"  3^  °    9  9      ^^  there  are  12  times'  8  eightlis,  or  ^^ ;  and  in  12  +  f 
"¥"  "   t  ~  ~t''     there  are  ^/-  +  f ,  or  -\^-.     Hence,  12|  is  equal  to  -^/. 

149.  From  the  above  solution  it  is  seen  that : 

In  changing  a  mixed  number  to  a  fraction,  the  integer  is 
multiplied  by  the  given  denominator,  to  this  loroduct  the 
numerator  of  the  fractioncd  part  is  added,  and  the  result  is 
ivritten  over  the  given  denominator. 

Keduce  the  following  to  improper  fractions  : 

2.  6/^.  7.    13f.  12.    36^.  17.    53^. 

3.  9|.  8.    18|.  13.    44^^,.  18.    651^.  • 

4.  7f.  9.    21^^.  14     352V  19.    74|f.    • 

5.  83^.  10=    32-j7-  15.    37if.  20.    87f|. 

6.  9^.  11.    18^-^  16,   2%  21.    98i|. 


152  ELEMENTS   OF   ARITHMETIC. 

150.   To   reduce   improper  fractions  to   integers   or  mixed 
numbers. 

1.  To  liow  many  oranges  are  4  half  oranges  equal?     6 
half  oranges  ?     8  half  oranges  ?     5  half  oranges  ? 

2.  To  how  many  dollars  are  4  quarter  dollars  equal  ?     8 
quarter  dollars  ?     16  quarter  dollars  ?     5  quarter  dollars  ? 

3.  In  10  half  bushels  how  many  bushels  are  there  ?     In 
12  half  bushels  ?     In  20  half  bushels  ? 

4.  How  many  units  are  there  in  3  thirds  ?     In  6  thirds  ? 
In  12  thirds  ?     In  15  thirds  ?     In  7  thirds  ? 

5.  How  many  units  are  there  in  f  ?    f?    ^?    i^?    lA? 
S.    How  many  yards  are  there  in  f  yards  ?     -i^  yards  ? 

^82- yards?     J^  yards?     -L^- yards  ?     ^5_  y^rds  ? 

7.    How  many  units  are  there  in  |  ?    |  ?    i£-?    i-i  9    u,  9 

1_I  9      13.9      lA  9       LS.  ?       13-9       S_§.  ?       SJ.  ?       4J.  ?       4J.  ? 
6'  2*  3'  3*  4'  3*  7*  8*  9* 

WRITTEN   EXERCISES. 

151o   1.    Eeduce  J^  to  a  mixed  number. 

i|-^  =  125 -T- T  =  17-f-.         Explanation. — Since    7    seventh*   are 
equal  to  1  unit,  125  seventlis  are  equal  to 
as  many  units  as  7  seventlis  are  contained  times  in  125  sevenths,  or 
17 f  units.     Therefore,  ^^  =  17f. 

152.   From  the  above  solution,  it  is  seen  that : 

In  reducing  an  imj)roper  fraction  to  a  mixed  number  the 
numerator  is  divided  by  the  denominator. 

Eeduce  to  integers  or  mixed  numbers. 


2. 

1A8. 

7. 

w- 

12. 

W- 

17. 

w- 

3. 

2^£ 

8. 

w- 

13. 

¥^. 

18. 

m- 

4. 

^K 

9. 

¥^ 

14. 

4G   * 

19. 

H^- 

5. 

L|i. 

10. 

W-- 

15. 

W- 

20. 

^m- 

6, 

¥/• 

11. 

899 
44  • 

J6. 

w- 

21. 

m^ 

FRACTIONS.  153 

153.  To  reduce  fractions  to  smaller  or  lower  terms. 

1.  Ill  1  of  a  yard  how  many  fourths  of  a  yard  are  there  ? 
How  many  sixths  ?     Eighths  ?     Tenths  ? 

2.  In  -1-  of  an  orange  how  many  sixths  are  there  ? 
How  many  ninths  ?     Twelfths  ? 

3.  How  are  the  terms  of  the  fraction  J  obtained  from 
those  of  i  ?     f  from  i  ? 

4.  Since  the  terms  of  the  fraction  |  are  obtained  from 
those  of  ^  by  multiplying  them  by  2,  how  may  the  terms  of 
1  be  obtained  from  those  of  J  ? 

5.  Since  the  terms  of  the  fraction  |  are  obtained  from 
those  of  i  by  multiplying  them  by  3,  how  may  the  terms 
of  ^  be  obtained  from  those  of  |  ? 

154.  From  these  examples,  it  is  seen  that  : 

The  value  of  a  fraction  is  not  changed  by  multiplying  or 
dividing  both  terms  by  the  same  number. 

6.  Change  to  twelfths :  i;  i;  |;  i;  |;  i;  f 

\  7.    Change  to  twentieths :  i;  J;  f;  i;  |;  tVJ   iV     * 
*   8.    Change  f  to  halves ;  f  to  fourths  ;  y\  to  thirds.    • 
9.    Change  to  thirds:  f ;  |;  f ;  f ;  ^5  H^  HJ  M;  M^ 

'  10.  Change  to  fifths:  3%;  3-^;  A;  A;  A;  Mj  fi^  if 

155.  A  number  that  is  an  exact  divisor  of  two  or  more 
numbers  is  a  Common  Divisor  of  those  numbers. 

156.  A  fraction  is  expressed  in  its  Smallest  or  Lowest 
Terms  when  its  numerator  and  denominator  have  no  common 

divisor. 

WRITTEN    EXERCISES. 

157.  1.    Reduce  J|  to  its  smallest  terms. 

g\4S       (*)  ExTLANATiON.  —  Siiice  the  fractioii  is  to  be  Teduced  to 

qV7^\  ~  ^  its  smallest  terms,  we  divide  the  terms  of  the  fraction  by 

'            '^  8,  and  the  terms  of  the  resulting  fraction  by  2.     The  terms 

'^)^^  _  'J!  of   the  fraction   |   have   no   common  divisor  ;  therefore 

2)^      "^  li»  when  reduced  to  its  smallest  terms,  is  equal  to  |. 


154  *  ELEMENTS   OF   ARITHMETIC. 

158.  It  is  evident,  therefore,  that : 

In  reducing  a  fraction  to  its  smallest  terms,  the  numerator 
and  denominator  are  divided  by  any  common  divisor,  and  this 
process  is  continued  until  the  terms  have  no  common  divisor^ 

Eeduce  to  their  smallest  terms  : 

2   18.       8   AA      14   _44_      20   -5-4- 

/5.   3g.        O.   gg.       i'±.   ;^QQ.       /CU.   j4^. 
^15.         q       lA  if;    17        oi   i6  0 

A        14        10   32        ifi    84        oo   150 

^'   6^-       ^1*  "TS*       •'-'''•  TST-       2^*  Tsf- 

Cl2  iol5  1Q         60  94        350 

b.      3-2-.  1/J.      3^.  10.      -240-  /i^-      49-0- 

721  I'^-'i'^  1Q  2  5  OK  375 

'•      6&-  ■L'5'      T5-  •^^-     T2T°  ^^-      10  5""0- 

159.  To  reduce  dissimilar  to  similar  fractions. 

1.  If  I  bought  1  of  a  pound  of  one  kind  of  tea,  and  i  of 
a  pound  of  another  kind,  how  many  sixths  of  a  pound  of 
each  kind  did  I  buy  ? 

2.  Mr.  Roe  planted  i  of  an  acre  with  potatoes,  and  i  of 
an  acre  with  corn.  How  many  tenths  of  an  acre  did  he 
plant-  with  each  ? 

3.  Mr.  Lee  has  -|  of  a  barrel  of  russet  apples  and  J  of  a 
barrel  of  Baldwin  apples.  How  many  twelfths  of  a  barrel 
has  he  of  each  ? 

4.  Change  |  and  ^  to  fractions  having  their  denominators 
alike,  i  and  i  i  and  i-.  i  and  i  i  and  i.  i  and  i 
h  h  and  i  i  1.  and  i.  J-,  i  and  ^V-  h  h  an^i  iV-  i.  h 
and  1.     i,  1,  and  1. 

160.  T'ractions  that  have  the  same  denominators  are 
called  Similar  Fractions. 

161.  Fractions  that  have  not  the  same  denomniators  are 
called  Dissimilar  Fractions^ 

162.  The  denominator  of  simikir  fractions  is  called  a 
Common  Denominator^ 


FRACTIONS.  155 

163.  When  similar  fractions  are  expressed  in  their  lowest 
terms,  they  have  their  Least  Common  Denominator. 

164.  1.    Eeduce  J,  |,  and  |i  to  similar  fractions. 

Explanation.  —  Since  the  fractions  are  to 
be  changed  to  other  fractions  having  a  common 
denominator,  the  terms  of  each  fraction  must 
be  multiplied  by  some  number  which  will  cause 
them  to  have  the  same  denominator. 

By  examining  the  denominators,  4,  8,  and  12, 
it  is  evident  that  the  denominators  of  all  the 
fractions  can  be  made  24,  and  the  fractions  will 
then  be  similar.  To  make  the  denominators  24,  the  terms  of  the  first 
fraction  must  be  multiplied  by  6,  the  terms  of  the  second  by  3,  the 
terms  of  the  third  by  2.  And  thus  the  fractions  are  changed  to  the 
similar  fractions,  if,  if,  f f . 


3  3x6 

4  4x6" 

18 
24* 

5       5x3 

8       8x3 

15 
"24' 

1      11x2 

22 

2      12x2 

24 

Eeduce  to  similar  fractions  : 


2. 

h  h  A-               6. 

3. 

h  h  tV         7. 

4. 

h  T%  A-             8. 

5. 

i  T%  sV             9- 

4. 

Eeduce  |,  f,  ^, 

10. 


h 

t  h  A- 

h 

h  ^,  tV- 

1 

8"' 

T2'  -iv  -iS- 

f- 

TT>   2T'  Y- 

3  2       3 
4'    5'    lo- 
ss      4  11 
6"?  TO?    5-  ■'•■'•• 

5.    _6_     3.  12 

8'    12'    4*  -^'^' 

4  _3_    13  lo 

75    21?  42-  -^^• 

and  ^  to  similar  fractions  having 
their  least  common  denominator. 

Explanation.  —  The  least  common  de- 
nominator cannot  always  be  easily  found 
by  inspection.  It  may  then  be  found  as 
in  the  margin. 

1114  Since   the   least  common  denominator 

3x2x2x4  =  48.  must  be  the  smallest  number  that  will  con- 
tain each  of  the  denominators,  it  must 
contain  each  of  the  prime  factors  of  the  denominators  and  no  other 
factors.  The  prime  factors  are  found  as  in  the  margin.  3  is  a  prime 
factor  of  3  and  12,  and  consequently  a  factor  of  the  least  common 
denominator.  Dividing  by  3,  and  writing  below,  the  quotients  and  num- 
bers of  which  3  is  not  a  factor,  we  have  1,4,  4,  16.  Dividing  by  2,  and 
again  by  2,  the  factors  of  the  denominator  are  found  to  be  the  divisors 
3,  2,  2,  and  the  factor  4  in  the  last  row.  Their  product  is  48,  the  least 
common  denominator.     The  fractions  thus  become  ||,  |f ,  |f ,  f|. 


3 

3  4  12  16 

2 

1  4     4  16 

2 

12     2     8 

156  ELEMENTS   OF   ARITHMETIC. 

165.  Froiii  these  examples  it  is  evident  that : 

In  reducing  dissimilar  to  similar  fractioyis  the  least  common 
denomincdor  is  found  by  multiplying  together  all  the  priyne 
factors  of  the  given  denominators.  The  terms  of  ectch  frac- 
tion are  then  midtiplied  by  such  a  member  as  icill  cause  each 
fraction  to  have  the  least  common  denominator. 

1.  In  finding  the  product  of  the  different  prime  factors,  each  fac- 
tor must  be  used  the  greatest  number  of  times  that  it  occurs  in  any 
of  the  given  numbers, 

2.  Reduce  all  mixed  numbers  to  improper  fractions. 

Eeduce  to  similar  fractions  having  the  least  common 
denominator : 

4  3.       _7_  on        4.4         3        23 

_2_j4j7  97^11413 

fi.       -4__      _7_  OQ        K1      az       2  0 

5>       125     3  0^-  ^O.      O3,    O3-,     gg-. 

A.      _J_       0.  OQ       07      43      3  1 

y       3  6>       5-  ^^-      ^9>    T2'J    T'S' 

ADDITION    OP    FRACTIONS. 

166.  1.  John  walked  i  of  a  mile  on  Monday,  |  of  a  mile 
on  Tuesday,  and  f  of  a  mile  on  Wednesday.  How  many 
eighths  of  a  mile  did  he  walk  in  all  ? 

2.  How  many  sixths  are  -f,  f,  and  1  ? 

3.  How  many  ninths  are  |-,  i,  and  I  ? 

4.  How  many  twelfths  are  ^,  yVj  ^.nd  -f^  ? 

5       4i2_9     3     I    1—9     JL-1_JL  — 9     -5__1_-J_— ?     2_|_3_9 
O-      7-1-7—-       ^-Ts—  •        10^10—-        ll'll  —  •       4'4—  • 

5.  J_  3  _  9      _5_  -1-  J7_  =  9       i  0     .    JL  —  9 

6T^6 13^^13  —  •  3U'30   —  ' 

6=  A  boy  spent  1  of  a  dollar  for  a  book,  and  i  of  •'^  dollar 
for  a  cap.     What  part  of  a  dollar  did  both  cost  ? 

7  A  farmer  sold  I  of  his  sheep  at  one  time  and  ^  of  them 
at  another.     What  part  of  his  sheep  did  he  sell  ? 

8.  How  many  sixths  are  there  in  1  and  i  ? 

9.  How  many  ninths  are  there  in  J  and  1  ? 


15. 

i 

A. 

9 
To- 

20. 

16. 

s 

8> 

f, 

T^e- 

21. 

17» 

1. 

A. 

3^. 

22. 

18. 

i, 

f. 

2V 

23. 

19. 

i, 

*, 

tV. 

24. 

FRACTIONS.  157 

10.  How  many  eighths  are  there  in  ^  and  i  ? 

11.  How  many  tenths  are  there  in  ^,  i,  and  -^  ? 

12.  What  kind  of  fractions  can  be  added  without  chang» 
ing  their  form  ? 

13.  What  must  be  done  to  dissimilar  fractions  before 
they  can  be  added  ? 

14.  Add  J  and  i;  i  and  f ;  J  and  |;  ^  and  |- ;  |-  and  i; 
■!•  and  |- ;  ^  and  ^^2 ;  i  and  -^ ;  |  and  -^\. 

15.  Add  i  and  i ;  i  and  i ;  ^  and  l- ;  i  and  -J- ;  i  and  i. 

16.  Add  1  and  | ;  i  and  f ;  |  and  | ;  f  and  |- ;  f  and  |-. 

17.  I  paid  f  i  for  milk  and  $  Jq- for  lettuce.  How  much 
did  I  pay  for  both  ? 

18.  James  paid  $4  for  a  fishing-rod  and  $yL-  for  a  line. 
What  part  of  a  dollar  did  both  cost  him  ? 

19.  Lucy  worked  ^  of  a  day  at  her  lessons  and  spent  i 
of  a  day  at  a  picnic.  What  part  of  the  day  did  she  spend 
at  both  ? 

20.  A  farmer's  son  sold  ^  dozen  eggs  to  one  nj.an,  J  dozen 
to  another,  and  f  dozen  to  another.  How  many  dozen  did 
he  sell  in  all  ? 

21.  A  lad  was  presented  with  some  money  on  his  birth- 
day. The  next  day  he  spent  4  of  it,  the  day  following,  i  of 
it,  and  on  the  third  day  he  spent  -^  of  it.  W^hat  part  of 
the  money  did  he  spend  ? 

WRITTEN    EXERCISES. 
167.   1.  What  is  the  sum  of  |,  f,  and  |? 

A  =  2.5.  ExpLAXATiox.  —  Siiice  the  fractions  are  not  similar, 

they  must  be  changed  to  similar  fractions  before  they 


are  added. 


3  3. 0. 

4  —  40 
4^ g^  2. 

5  —  40^  The  least  common  denominator  of  the  given  frac- 

8  7  —  o  7_     tions  is  40,  and  4  =  M,  1  =  M,  '"uitl  ^  =  32      Hence 
the  sum  of  the  given  fractions  will  be  the  sum  of  |f , 
f§,  and  f-2,  which  is  f^,  or  2^^. 


158  ELEMENTS  OF   ARITHMETIC. 

168.   From  the  above  solutions,  it  is  evident  that : 

In  adding  fractions,  dissimilar  fractions  are  first  changed  to 

similar  ones,  then  their  numerators  are  added,  and  the  sum 

is  placed  over  the  common  denominator. 

1.  If  the  sum  is  an  improper  fraction,  it  should  be  reduced  to  an 
integer  or  mixed  number. 

2.  When  there  are  mixed  numbers  or  integers  to  be  added,  the 
integers  and  fractions  should  be  added  separately  and  then  those 
results  added. 

Find  the  sum  of  the  following  : 

10.  6|,  2|,  and  4^^. 

11.  ^,  1^-^,  and  ^. 

12.  4J,  3f,  and  8^V 

13.  5|,  101^,  and  ii. 

14.  6f,  4f,'^andl2V 

15.  121,  H^  ^^^  h\' 

16.  9^%  i,  and  llij. 

17.  Ti^,  3U,  and  6ii. 

18.  A  farmer  sold  a  load  of  hay  for  $8f,  a  load  of  oats 
for  $  22|,  and  a  load  of  wheat  for  $  53|.  How  much  did  he 
receive  for  all  ? 

19.  James  picked  5|  bushels  of  apples  from  one  tree, 
7^  bushels  from  another,  6j\  bushels  from  another,  and  8|- 
bushels  from  another.  How  many  bushels  did  he  pick 
in  all  ? 

20.  Three  barrels  of  sugar  contained  respectively  225| 
pounds,  232y''2-  pounds,  and  240^-  pounds.  What  was  the 
weight  of  the  whole  ? 

21  o  A  school-room  is  32|  feet  long  and  29 J  feet  wide. 
What  is  the  distance  around  the  room  ? 

22.  An  express  train  starts  from  Boston  at  IJ  p.m., 
reaches  Springfield  2J  hours  later,  and  Albany  4y^^  hours 
later  still.     At  what  o'clock- does  it  arrive  in  Albany  ? 


2. 

f. 

A. 

and  1^. 

3. 

A. 

A, 

and  Y^2-- 

4. 

f. 

f 

and  ^. 

5. 

h 

TO"' 

and  ii 

6. 

A, 

"2"0' 

and  |. 

7. 

f. 

2% 

and  y3^. 

8. 

f, 

A. 

and  |. 

9. 

A, 

^, 

and  I-. 

FRACTIONS.  159 


SUBTRACTION  OF  FRACTIONS. 

169.    1.  Bertie  earned  $|  and  spent  ^|.      How  much 
had  he  left  ? 

2.  Susie  had  |  of  a  pound  of  candy,  but  she  ate  f  of  a 
pound.     How  much  had  she  left  ? 

3.  There  were  f  of  a  bushel  of  oats  in  a  bin.  If  |  of  a 
bushel  was  taken  out,  how  much  remained  ? 

^•T         T—   •  12.         12—-  6  6—    •  10  10 

11  5     ?       11 8_  _  ?       J_8  _  i.0  —   '? 

16  16~~-         13  13   ~   '         25  25  ' 

5.  If  I  have  $1  and  give  away  $^,  Avhat  part  of  a  dol- 
lar shall  I  have  left  ? 

6.  A  man  who  had  J  of  a  cord  of  wood  sold  i  of  a  cord. 
What  part  of  a  cord  had  he  left  ? 

7.  One  pitcher  held  |  of  a  quart  of  milk,  and  another 
held  f  of  a  quart.  How  much  more  did  the  one  hold  than 
the  other  ? 

Q       1_1—  ?      3_i_9     i_i—  9     5._1  — *?     l  —  L—9 
°-2"  i  ~   '       4  2  —   •       2  G"~*       6  2  ~  •       2  8~* 

7._  i_  ? 

8  2  —   • 

9.  What  kind  of  fractions  can  be  subtracted  without 
changing  their  form  ? 

10.  What  must  be  done  to  dissimilar  fractions  before 
they  can  be  subtracted  ? 

11.  l-t=?   1-|  =  ?    l-f  =  ?    1-|  =  ?    1-^  =  ? 

19        1_1—  ?      1_1—  9     1_2_'?     3._3._9     2 5__? 

1^.^  8"—  -3  6  —  -3  9—   -4  8~-3  12—' 

11  _  5  _  9 

12  G  —  • 

io        1  1  9       1  2  9      2  1—9      4_2_9      5_1_9 

^^'      2:^3  —  •        2"~5"—  •        3  4—-       "5  3—-       6  4—   • 

7. 5_  _  9 

8  12  —  • 

14.  A  lad  earned  one  day  $f  and  spent  $|.  How  much 
had  he  left  ? 

15.  A  mechanic's  wages  were  $3  per  day,  but  he  paid 
$  I  per  day  for  his  board.     How  much  did  he  save  daily  ? 

16.  A  boy  agreed  to  work  for  a  man  5  days,  but  lost  If 
days  by  sickness.     How  many  days  did  he  work  ? 


160  ELEMENTS   OF   ARITHMETIC. 

17.  A  boy  raised  vegetables  which  he  sold  for  $3,  The 
cost  of  seed  and  other  expenses  were  ^lyV  How  much 
was  his  net  profit  ? 

18.  A  man  bought  a  hat  for  $2i  a  vest  for  $2|,  and  a 
linen  coat  for  $  li.  How  much  change  should  he  receive 
from  the  merchant,  if  he  paid  for  them  with  a  ten-dollar 
bill  ? 

WRITTEN    EXERCISES. 

170.  1.  What  is  the  difference  between  ^  and  ^2"  ^ 

Explanation.  —  Since  the  fractions  are  not  similar,  they 
JLA  __.  4  5.      must  be  made  similar  before  they  can  be  subtracted.    The 

^  least  common  denominator  of  the  given  fractions  is  48  ; 
T 2^  ~  15      i|  =  |f,  and  y7^  =  ||.     Hence,  the  difference  between  the 

^      given  fractions  is  the  difference  between  ||  and  ||,  which 

io    1  7 
IS  5g. 

2.    What  is  the  difference  between  15|  and  8f  ? 

^p.„       ir9   1121  Explanation. — Since    the    numbers   are 

■f  ~    ''^TT  —       T2"     composed  of  integers  and  fractions,  each  may 

8|  =    ^rf  ~    ^1^2"     ^^  subtracted  separately.     Reducing  the  frac- 

(^ij.     tions  to  similar  fractions,  it  is  evident  that  i| 

^  ^      cannot  be  subtracted  from  y\,  hence  1,  taken 

from  15,  is  united  with  j%.     1,  or  i|,+  j%  =  fi.    Subtracting  81f  from 

142  1,  there  is  a  remainder  of  Oil. 

171.  From  the  solution  of  the  above  examples  it  is  seen 

that : 

In  subtracting  fractions,  dissi7nila7'  fractions  are  changed  to 
similar  ones,  then  their  7iiimerators  are  subtracted,  and  the 
difference  is  ivritteri  over  the  common  denominator. 

When  there  are  integers  or  mixed  numbers  to  be  subtracted,  the 
integers  and  fractions  should  be  subtracted  separately. 


3. 

4. 

5. 

6. 

7. 

8. 

9. 

Out  of 

H 

5 

T 

fV 

T% 

if 

4t 

H 

Take 

1 

1 

A 

A 

A 

A 

H 

FRACTIONS. 

161 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

From 

A 

3\ 

il 

9 

20 

15 

26 

Take 

t\ 

A 

H 

2t\ 

'A 

1<^A 

'H 

17. 

18. 

19. 

20. 

21. 

22. 

23. 

From 

H 

8A 

m 

IGf 

20t\ 

31tV 

63t\ 

Take 

2f 

2A 

5| 

lOf 

16if 

20|t 

39ti 

24.  A  box  of  soap  weighed  75|  pounds.  Tlie  weight  of  tlie 
box  alone  was  3^^  pounds.    How  much  did  the  soap  weigh  ? 

25.  From  a  farm  of  120  acres  there  were  sold  36Y^g  acres. 
How  much  was  left  ? 

26.  A  flag-staff  50^^  feet  high  was  broken  off  by  a 
storm,  so  that  it  measured  43|-  feet.  How  much  was 
broken  off  ? 

27.  A  grocer  had  128|-  pounds  of  sugar  in  one  barrel,  and 
98|  pounds  in  another.  How  much  more  was  in  the  one 
barrel  than  the  other  ? 

28.  A  clerk  earned  $  75  per  month.  His  expenses  during 
that  time  were  as  follows:  board,  $22i-;  washing,  $5|; 
and  other  expenses  $16|.  How  much  did  he  save  per 
month  ?  ' 

29.  A  man  spent  -^^  of  his  money  for  a  house,  -^  for 
furniture,  and  |  for  horses  and  carriages.  What  part  of  liis 
money  had  he  left  ? 

30.  The  sum  of  two  numbers  is  llfV  One  of  the  num- 
bers is  8J.     What  is  the  other  number  ? 

31.  From  a  web  of  cotton  cloth  containing  58j^g-  yards, 
8J  yards  were  cut  off  at  one  time,  and  15f  yards  at  another 
time.     How  many  yards  remained  ? 

32.  Mr.  Allen  left  f  of  his  property  to  his  wife,  -^  to  his 
sons,  i  to  his  daughter,  and  the  remainder  to  charitable 
institutions.  What  part  of  his  property  was  left  to  chari- 
table institutions  ? 

EL.    OF    AR,  —  11 


162  ELEMENTS  OF  ARITHMETIC. 

Find  the  value  of  the  following : 

33.  f  +  A  +  M  +  A-  40-  3i  +  4|  +  3f-lf 

34-  l+l  +  A  +  H-  41.  6f-4f  +  3TV-lfV. 

35-  |  +  T'«-i-i-  42.  8t  +  2J^-3i-3|. 

36-  A  +  A-A-sV  43.  2|  +  3|-lf  +  2|. 
37.  A-A  +  f-A-  44.  6i-2f-l|  +  4A. 

Qft       _5 3_     I     6  _  1  5  4.K       93     I    A  5    Q  3    14 

^°-      2  1  42^    IT        "84-  ^^'     ^  5"  +  "TT        '^TO         -'-T5"- 

39       i 5.  _  2.    I     3. _7_  4fi        A5     I     23_15     I     94 

^^•16  3^4  12-  ^'^-      '^T  ^  ^4         -•-¥  +  ^9- 

MULTIPLICATION  OF  FRACTIONS. 
172.   To  multiply  a  fraction  by  an  integer. 

1 .  At  $  I"  a  yard,  what  will  2  yards  of  cloth  cost  ? 

2.  If  a  family  uses  f  of  a  pound  of  butter  each  day, 
how  much  will  it  use  in  2  days  ? 

3.  If  a  man  earns  ^^^  an  hour,  how  much  will  he  earn 
in  3  hours  ? 

4.  How  much  is  3  times  |- ?     4  times  |?     5  times  -^? 
3  times  /q  ?     2  times  y4_  ?     7  times  y2_  ?     6  times  ^\  ? 

6.    How,  then,  may  a  fraction  be  multiplied  by  any 
integer  ? 

6.  If  I  gave  i  of  an  orange  to  each  of  5  boys,  how  many 
halves  did  I  give  away  ?  How  many  whole  oranges  Avas  that  ? 

7.  At  $1  a  yard,  what  will  2  yards  of  velvet  ribbon 
cost? 

8.  Mary's  mother  has  4  tumblers  of  jelly,  each  holdiug 
I  of  a  pint.     How  many  pints  of  jelly  has  she  ? 

9.  5  times  |=?  2  times  |=?  3  times  |=?  4  times -|=? 
2  times  f=?  3  times  |  =  ?  4  times  ^=?  6  times  ^  =  ? 

10.  How  many  are  4  times  1|  ?  3  times  2|  ?  5  times  3J  ? 

11.  Multiply  4f  by  5 ;  5A  by  3;  94  by  8 ;  6f  by  9. 

12.  How  much  is  5  times  y^?     Express  the  result  in  its 
smallest  terms.    How  may  this  result  be  obtained  from  y*^-  ? 


FRACTIONS.  163 

13.  How  much  is  3  times  yL?      Express  the  result  in 
its  smallest  terms.      How  may  this  result  be  obtained  from 

14.  In  what  other  way,  therefore,  may  we  -sometimes 
multiply  a  fraction  by  an  integer  ?     Give  both  ways. 

15.  If   a  bookseller  sells  15  books  at  ^^^  apiece,  how 
much  will  he  get  for  them  ? 

16.  When  flour  sells  at  ^4-J  per  barrel,  how  much  will  9 
barrels  cost  ? 

17.  A  clerk's  wages  were  $101  per  week.     How  much 
would  he  earn  at  that  rate  in  8  weeks  ? 

18.  If  the  wages  of  a  laborer  are  $  If  per  day,  how  much 
does  he  earn  in  10  days  ? 

19.  A  good  walker  can  walk  4^  miles  per  hour.     How 
far  can  he  walk  at  the  same  rate  in  8  hours  ? 

173.   It  has  been  discovered,  therefore,  that : 
A  fraction  can  he  multiplied  by  an  integer  by  multiplying 
the  numerator  or  dividing  the  denominator  by  the  integer. 

WRITTEN   EXERCISES. 


174.   1.   Mult: 

Lpiy 

«by8. 

'■ 

24^^-     24     • 

_88 
24 

Explanation.  — 8  times 

=  3if,or3f. 

iiisff,  or  3i|,  or3|.  Or, 
since  dividing  the  denom- 

«■•'   i-'  =  24 

11 

.-8 

11        OO 

inator  multiplies  the  frac- 
tion, 8  times  ^\  is  -y-,  or 
3|. 

Multiply : 

3 

2.   Aby5. 

7.      |byl6. 

12.      3f    by    6. 

3.  A  by  4. 

8.  yVbyi5. 

13.      Gf    by  12. 

4.   iXby6. 

9.    11  by  14. 

14.    10^^  by  15. 

5.    il  by  5. 

10.    j§5by20. 

15.    14f    by  18. 

6.  H  by  a 

11.    A  by  23. 

16.    le^by    8, 

164  ELEMENTS  OF  ARITHMETIC. 

17.  What  will  5  pounds  of  cheese  cost  at  14f  ^  per  pound  ? 

18.  If  a  train  goes  at  the  rate  of  20|  miles  an  hour,  hov/ 
far  will  it  go  in  12  hours  ? 

19.  When  potatoes  are  worth  66|^  a  bushel,  what  must 
I  pay  for  20  bushels  ? 

20 o    How  much  will  15  tons  of  coal  cost  at  $  6^^^  a  ton  ? 

21.  How  many  feet  are  there  in  10  rods,  since  there  are 
161  feet  in  1  rod  ? 

22.  A  boy  can  chop  if  of  a  cord  of  wood  a  day.     How 
much  can  he  choj)  in  3  weeks,  or  18  days  ? 

23.  If  a  man  works  Sf  hours  per  day,  how  many  hours 
will  he  work  in  20  days  ? 

24.  If  the  railroad  fare  between  two  places  is  $  6|-^,  what 
will  the  fare  of  10  persons  cost  ? 

175.   To  multiply  an  integer  by  a  fraction. 

1.  James  had  12  chickens,  but  sold  i  of  them.  How 
many  did  he  sell  ?     What  is  f  of  12  chickens  ? 

2.  What  will  ^  of  a  pound  of  raisins  cost  at  16^  per 
pound  ?     What  is  f  of  16^  ? 

3.  Mr.  Bird's  horse  can  trot  5  miles  in  20  minutes. 
How  many  minutes  will  it  take  him  to  go  1  mile  ?  What 
is  I  of  20  ?    I  of  20  ?     I  of  20  ? 

4.  What  is  f  of  24?     f  of  14  ?     |  of  18  ?     f  of  28  ? 

5.  Two  yards  of  ribbon  were  divided  equally  among  3 
girls.     How  much  did  each  girl  get  ? 

6.  If  1  yard  of  braid  is  worth  3^,  what  is  J  of  a  yard 
worth  ?     What  is  f  of  4^  ? 

7.  Four  quarts  of  milk  filled  5  pitchers  of  equal  size. 
How  much  did  each  pitcher  hold  ?     What  is  |  of  4  ?     f  of 

4  ?     I  of  4  ? 

8.  What  is  i  of  3  ?    J-  of  2  •?   tV  of  7  ?   -J  of  3  ?    i  of  5  ? 

9.  What  is  1  of  4  ?     I  of  4  ?     |  of  7  ?     i  of  5  ?     J  of 

5  ?    I  of  9  ?    i  of  8  ?    f  of  8  ?    I  of  15  ?    i  of  12  ?    |  of  12  ? 


FRACTIONS.  165 

10.  A  coat  cost  $  12^  and  a  x^air  of  trousers  and  a  vest  J 
as  much.     How  much  did  the  suit  cost  ? 

11.  A  schoolboy  paid  yL  of  his  money  for  a  pencil,  f  of 
it  for  a  grammar,  and  i  of  it  for  a  penholder.  If  he  had 
40^  before  he  made  his  purchases,  how  much  had  he  left  ? 

12.  A  lesson  in  arithmetic  contained  30  examples.  Henry 
solved  -f  of  them,  and  his  sister  solved  f^  of  them.  How 
many  did  each  solve  ? 

13.  The  month  of  February  has  usually  28  days.  If,  in 
1891,  ^  of  them  were  stormy,  how  many  were  pleasant  ? 

14.  A  paymaster  paid  his  men  only  4  of  their  monthly 
wages.  If  each  man  earned  $  35,  how  much  did  he  pay  to 
each  ? 

176.  From  these  examples  it  is  seen  that : 
Multiplying  a  number  by  a  /inaction  is  finding  a  fractional 

part  of  the  number. 

Thus,  to  multiply  12  by  f  we  simply  find  f  of  12. 

WRITTEN    EXERCISES. 

177.  1.    Multiply  13  by  f,  or  find  I- of  13. 

ExPLANATiox.  —  To    multiply 

13  X  -  =  ^^  ^  "^  =  —    or  101.      ^^  ^y  t  '3  to  lind  4  of  13.     i  of  13 

5  5  5  '  ^       is  y,  and  |  of  13  is  4  times  ^3^, 

which  is  -\2_^  or  lOf . 
When  the  multiplier  is  a  mixed  number,  multiply  by  the  integer 
and  fraction  separately,  and  add  the  results. 


What  is : 

Multiply : 

jVIultiply : 

2.      f  of  15  ? 

8.    54byx 

14.    ISbylJ. 

3.      fofl8? 

9.    35by|. 

15.    25  by  3f. 

4.    ^  of  34  ? 

10.    48by|i. 

16.    42by5f. 

5.      |of24? 

11.    64  by  If. 

17.    34  by  71 

6.    ^oi2S? 

12.    81byH. 

18.    29by|. 

1.   ^of45? 

13,    60  by  2V 

19.    m  by  9f 

166  ELEMENTS   OF   ARITHMETIC. 

20.  Wlien  corn  is  worth  75^  a  bushel,  what  is  the  value 
of  f  of  a  bushel  ? 

21.  There  are  125  acres  in  Mr.  Swift's  farm.  How  many 
acres  are  there  in  |-  of  the  farm  ? 

22.  The  expenses  of  a  foundry  for  one  month  were  $  583, 
What  will  f  of  the  expenses  be  ? 

23.  What  will  6f  yards  of  silk  cost  at  88^  a  yard  ? 

24.  The  average  rate  per  hour  of  a  very  fast  express 
train  is  52  miles.     How  far  will  it  go  in  S^q  hours  ? 

25.  What  will  be  the  cost  of  8^  pounds  of  tea  at  30j^  a 
pound,  6^  pounds  of  sugar  at  5^  a  pound,  and  13^  pounds  of 
raisins  at  16^^  a  pound  ? 

26.  What  will  be  the  cost  of  6|  dozen  arithmetics  at  $8 
per  dozen,  5^  dozen  grammars  at  $6  per  dozen,  and  6|- 
dozen  geographies  at  $  9  per  dozen  ? 

178.   To  multiply  a  fraction  by  a  fraction. 

1.  If  i  of  an  apple  is  divided  into  2  equal  parts,  what 
part  of  the  apple  will  each  part  be  ?  How  much  is  i  of  -J 
of  an  apple  ? 

2.  If  ^  of  a  yard  is  divided  iiito  2  equal  parts,  what 
part  of  a  yard  will  each  part  be  ?     How  much  is  i  of  J  ? 

3.  If  I  of  a  pie  is  divided  into  2  equal  parts,  what  part 
of  the  pie  will  each  part  be  ?     How  much  is  i  of  ^  ? 

4.  If  1  of  a  pie  is  divided  into  3  equal  parts,  what  part 
of  the  pie  will  each  part  be  ?     How  much  is  -i-  of  |-  ? 

5.  How  much  is  1  of  1?     ^  of  i  ?     i  of  i  ?     ^ofi? 
i  of  1  *?      i  of  1  '^ 

Since  -i-  of  i  is  -^^,  how  much 

7.  A  man  having  i  of  an  acre  of  land  sold  f  of  it.  What 
part  of  an  acre  did  he  sell  ? 

8.  If  a  yard  of  crape  costs  $  J,  what  part  of  a  dollar 
will  -|  of  a  yard  cost  ? 


i 

on?  iofi? 

6.    How  much 

iofi? 
isiofi? 

is 

fofi?     fofi 

?     fofi? 

FRACTIONS.  167 

9.    I  bought  |-  of  a  bushel  of  beans  and  planted  f  of  them. 
What  part  of  a.  bushel  did  I  plant  ? 

10.  How  much  is  |^  of  i  ?  Since  i  of  -J-  is  yL^  how  much 
isiof-l?     ioff?     ioff?     foff?     |of|? 

11.  When  cloth  is  worth  $|-  a  yard,  what  will  J  of  a  yard 
cost  ?     What  will  f  of  a  yard  cost  ? 

12.  Mr.  Stone,  having  a  lot  containing  ^  of  an  acre,  sold 
i  of  it.  What  part  of  an  acre  did  he  sell  ?  If  he  had  sold 
-|  of  it,  what  part  of  an  acre  would  he  have  sold  ? 

13.  If  a  boy  who  had  ^i  spent  i  of  it  for  candy,  what 
part  of  a  dollar  did  he  spend  ?  If  he  had  spent  |  of  it, 
what  part  of  a  dollar  would  he  have  spent  ? 

14.  Mr.  Stearns,  who  owned  |  of  a  store,  sold  f  of  his 
share.     What  part  of  the  whole  store  did  he  sell  ? 

15.  What  is  f  of  I?     fof^?     |of|?     foff? 

WRITTEN    EXERCISES. 

179.  1.  Multiply  |  by  f,  or  find  f  of  |. 

Explanation. — To  multiply  |  by  f,  is  to 
?  X  -  =  ^  ^^  =  A      find  f  of  f.     }  of  I  is  ^V  (yJ-J;   then,  ^  of  f 
8      7      ^  X  7      14     will  be  (y|,-g)  /^  ;  and  f  of  f  will  be  4  times 
2  -5%  .  or  It;  or  t\. 

180.  From  the  above  it  is  seen  that : 

'  In  multiplying  a  fraction  by  a  fraction,  the  numerators  are 
multiplied  together  for  the  numerator  of  the  p)roduct  and  the 
denominators,  for  its  denominator. 

1.  Reduce  any  mixed  numbers  to  improper  fractions. 

2.  Integers  may  be  expressed  as  fractions  by  writing  1  as  a  denomi- 
nator.   Thus,  4  may  be  written  f. 

3.  When  possible  use  cancellation. 

4.  The  word  of  between  the  fractions  is  equivalent  to  the  sign  of 
multiplication.  Such  expressions  are  sometimes  called  Compound 
Fractions.     Thus,  f  of  f  is  equal  to  f  x  f . 


168  ELEMENTS   OF   ARITHMETIC. 

Find : 

2.  J  off.  6.  if  off.  10.  -.Voff 

3.  A  off  7.  If  of||.  11.  i-VofrV 

4.  6  of  14.  8.  ^oi^.  12.  Aofi^. 

5.  fofff.  9.  HoffI-  13.  Ifofif. 

Find  :  Multiply : 

14.  I  of  41  18.    fofi-byf 

15.  foflOf  19.    fof^bySf. 

16.  -j-VofS^-V  20.    53  by  2  of  3^. 

17.  Hof7|.  21.    6fby|of4|. 

Find  the  value  of; 

22.  |x|X^Xf.  24.    IxlX^Xif 

23.  fxfxHxif.  25.    2-^x|xHxf. 

26.  There  are  16^  feet  in  a  rod.  How  many  feet  are 
there  in  J  of  a  rod  ?     In  3|-  rods  ? 

27.  When  hay  is  worth  $20 J  per  ton,  how  much  will  j^ 
of  a  ton  cost  ? 

28.  At  $  61-  a  ton,  what  must  I  pay  for  8|  tons  of  coal  ? 

29.  A  man  purchased  |  of  420|-  acres  of  land  and  then 
sold  I  of  what  he  had  bought.  How  many  acres  did  he 
sell?' 

30.  At  an  auction  sale  a  cow  was  sold  for  $  39|,  and  a 
horse  for  3|  times  as  much  as  the  cow.  For  how  much  was  ' 
the  horse  sold  ? 

31.  If  a  square  foot  of  land  is  Avorth  16|j^,  what  is  the 
value  of  lOf  square  feet  ? 

32.  The  cloth  of  Kuth's  dress  cost  ^^.  The  making 
cost  J  as  much,  the  trimming  cost  -J-^  as  much  as  tlie  making, 
and  the  linings  cost  4  as  much  as  the  trimmini^s.  What 
did  the  linings  cost  ?  What  was  the  whole  cost  of  the 
dress  ? 


FRACTIONS.  169 

DIVISION  OF  FRACTIONS. 
181.   To  divide  a  fraction  by  an  integer. 

1.  If  I"  of  an  orange  is  divided  equally  between  2  girls, 
what  part  of  the  orange  will  each  girl  have  ?    How^  much  is 

4  _j_2  ? 

2.  A  man  divided  |  of  an  acre  into  3  equal  lots.  How 
large  was  each  ?     How  much  is  |  -^  3  ? 

3.  If  a  horse  ate  -|  of  a  bushel  of  oats  in  4  days,  how 
much  did  he  eat  each  day  ?     How  much  is  f  -;-  4  ? 

4.  In  dividing  a  fraction  by  an  integer,  wdiat  part  of  the 
fraction  is  divided  ? 

5.  If  $  ^  is  divided  equally  between  2  boys,  what  part  of 
a  dollar  will  each  boy  get  ?     How  much  is  i  -r-  2  ? 

6.  When  3  pounds  of  sugar  can  be  bought  for  $  i,  what 
is  the  price  per  pound  ?     How  much  is  ^  -=-  3  ? 

7.  If  J  of  a  ship  is  owned  equally  by  5  men,  what  part 
of  the  ship  is  owned  by  each  ?     How  much  is  ^  -i-  5  ? 

8.  In  what  other  way,  besides  dividing  the  numerator, 
can  a  fraction  be  divided  by  an  integer  ?  State  both  ways 
in  which  a  fraction  may  be  divicled  by  an  integer. 

9.  If  3  dozen  oranges  can  be  bought  for  $4,  what  do 
they  cost  per  dozen  ? 

10.  If  -|  of  a  pound  of  tea  is  divided  equally  among  3 
persons,  what  part  of  a  pound  will  each  receive  ? 

11.  If  f  of  a  bushel  of  walnuts  fills  4  bags  of  equal  size, 
what  part  of  a  bushel  does  each  bag  hold  ? 


12. 

Mrs.  Jay  filled   5  tumblers   with  if  of  a  gallon  of 

jelly. 

How  much  did  each  tumbler  hold  ? 

13. 

Divide|by2;  f  by2;  ^byS;  tVl^V^;  if  by  5. 

14. 

Divide  f  by  2;  i  by  3;  f  by  4;  A  by  5;  -J  by  6. 

15. 

If  3  yards  of  cloth  cost  $  14,  what  does  1  yard  cost  ? 

16. 

When  5  bushels  of  potatoes  cost  34i,  what  does  a 

bushel  cost  ? 

170  ELEMENTS   OF   ARITHMETIC. 

17.  If  3  dozen  oranges  cost  $  |,  what  does  1  dozen  cost  ? 

18.  If  4  brooms  are  worth  $-j§g-,  how  much  are  they 
worth  apiece  ? 

19..  When  I  can  buy  4  dozen  eggs  for  ^^-^,  what  is  the 
price  per  dozen  ? 

20.  If  5  yards  of  gingham  sell  for  $^,  what  is  the  price 
per  yard  ? 

21.  If  6  pounds  of  tea  are  worth  $1|,  what  is  a  pound 
worth  ? 

22.  When  5  bushels  of  wheat  sell  for  $4i,  what  is  the 
price  per  bushel  ? 

23.  James  earned  $  6|  in  6  days.  What  were  his  daily 
wages  ? 

24.  A  boy  divided  6i-  dozen  marbles  equally  among  his  5 
sisters.     How  many  did  each  receive  ? 

182.  It  is  apparent  from  the  solution  of  the  above  exam- 
ples that : 

A  fraction  may  be  divided  by  an  integer  by  dividing  the 
numerator  or  multiplying  the  denominator  of  the  fraction  by 
the  integer.  * 

WRITTEN    EXERCISES. 

183.  1.  Divide  ||  by  5,  or  find  ^  of  f|. 

Explanation.  —  Since  dividing  the  nu- 

20  20  -^  5       4         nierator  of  a  fraction  divides  the  fraction, 

;tt  -T-  5  =  —   '       =  — -.       the  fraction  f f  may  be  divided  by  5  by 

^^  ''^        dividing  the  numerator  by  5.     Hence  the 

result  is  /y. 

2.    Divide  ^  by  6,  or  find  J  of  j\. 

Explanation.  —  Since  multiplying  the 

5  rj  !^  denominator  of  a  fraction  divides  the  frac- 

:r7;  -^  6  =  — — — -  =  ~.        tion,  the  fraction  j\  may  be  divided  by 

1^  Iw  X  0       i  ^        6  by  multiplying  the  denominator  by  C. 

Hence  the  result  is  y^^. 


Divide : 

-B  by  4. 


f^by5. 
«by8. 


bv 


FRACnONSc 

7.    If  by  12. 

11. 

ijby23. 

8.    If  by  16. 

12. 

If  ^J  28. 

9.    ffbylS. 

13. 

tf^yss. 

10.    If  by  15. 

14. 

if  by  42. 

171 


113  =  _4_7_ 


15.    Divide  llf  by  4. 

Explanation.  —  The  mixed  number  may 
be  reduced  to  an  improper  fraction,  and  tlie 
division  performed  as  before. 

Or,  tlie  number  may  be  divided  wittiout  be- 
ing reduced  to  an  improper  fraction.  Tlius,  4 
is  contained  in  11|  2  times,  and  a  remainder 
of  3|,  or  ^^- ;  and  -Ls  divided  by  4  equals  ^f. 
Hence  the  result  is  2 if. 


Or, 
4)111 


915 

^re- 


Divide  : 

16.  17|  by  7. 

17.  15^  by  4. 

18.  18f  by  5. 


22, 
23. 
24. 


125f  by  7. 


153|  by  8. 
200f  by  9. 


19.  25fby8. 

20.  31fby6. 

21.  46f  by  9. 

25.  Tlie  cost  of  5  tickets  from  Troy,  N.Y.,  to  Bethlehem, 
K.H.,  was  $  404.     How  much  did  each  cost  ? 

26.  The  rain  interfered  with  a  laborer's  working  31-  work- 
ing days  in  2  weeks.  How  many  days  did  he  work  on  an 
average  per  week  ? 

27.  A  farmer  forgot  the  price  which  he  received  for  his 
butter  per  pound,  but  he  knew  that  he  received  $2|  for  8 
pounds.     At  that  rate,  what  did  he  receive  per  pound  ? 

28.  An  estate  of  ^3956|  was  divided  equally  among  5 
heirs.     What  sum  did  each  receive  ? 

29.  The  earnings  of  5  men  for  10  days  were  $87|-.  What 
were  the  average  earnings  of  each  per  day  ? 

30.  If  a  man  can  reap  a  field  in  lOf  days,  in  what  time 
can  12  men  reap  it  ? 


172  ELEMENTS   OF   ARTTILAIETTC. 

31.  If  a  locomotive  runs  I'^^^q  miles  in  6  liours^  what  is 
the  average  rate  of  speed  per  hour  ? 

32.  A  field  containing  125|-  square  rods  was  divided  into 
10  lots  of  equal  size.     How  large  were  the  lots  ? 

184.  To  divide  an  integer  by  a  fraction. 

1.  How  many  tenths  of  a  dollar  are  there  in  a  dollar? 

2.  At  $  yig-  each,  how  many  quires  of  paper  can  be  bought 
for  $  1  ?     For  $  2  ?     For  f  3  ?     For  ^  4  ? 

3.  When  butter  is  worth  $  i  per  pound,  how  many  pounds 
can  be  bought  for  $1?     For  $  2  ?     For  $  3  ? 

4.  How  many  penknives,  at  $i  apiece,  can  be  bought  for 
$  1  ?  For  $  2  ?  How  many  at  $  f  apiece  can  be  bought 
for  $2?     For  {^ 4  ? 

5.  If  a  boy  spends  -|  of  an  hour  in  going  to  school,  in 
how  many  days  will  he  spend  3  hours  ?    6  hours  ?    9  hours  ? 

6.  A  man  gave  his  daughter  ^f  a  month  to  spend  as  she 
pleased.     In  how  many  months  would  he  give  her  $  9  ? 

7.  The  retail  price  of  one  kind  of  molasses  is  $  f  per 
gallon.     How  many  gallons  can  be  bought  for  $S? 

8.  A  young  man  spends  |  of  each  day  in  work  and 
recreation.  In  what  time  will  he  spend  12  days  in  this 
way  ? 

9.  How  many  times  is  i  contained  in  5  ?     In  9?    In  12? 

WRITTEN   EXERCISES. 

185.  1.  Divide  5  by  f,  or  find  how  many  times  f  is  con- 
tained in  5. 

q      ;-      rr  EXPLANATION.  —  Sliice  \  IS  contaliied 

5  -f —  =:  — ^ —  =  11|-.  in  1  7  times,  it  is  contained  in  5,  6  times 

'3  7  times,  or  85  times,  and  4  is  contained 

Or,  in  6   I  of  35  times,  or  llf  times.     Or, 

K 3  5.    35_5_3_ii2  sincB  in  5  there  are  Y,  >  is  contained  in 

^  -  T- J  -T-  •    7  -  ^^3-  3^5  111  times. 

1.  Cancellation  can  often  be  used  advantageously. 

2.  Mixed  numbers  may  be  reduced  to  improper  fractions. 


FRACTIONS. 

173 

Find  the  quotients  of : 

2. 

18  ^|. 

7.  47  --    f. 

12.  68- 

-H' 

17. 

225 -ff. 

3. 

21 -f 

8.  38 -3^^. 

13.   71 H 

.     1  2 
"  T3- 

18. 

387-^3. 

4. 

16 -f. 

9.   21-    f 

14.  43- 

^tV 

19. 

477^^. 

5. 

24  ^|. 

10.  36 --yV 

15.  29- 

■^H- 

20. 

642-1^. 

6. 

36 -|. 

11.  43 -A- 

16.  35- 

^if 

21. 

936 -if 

Divide  the  following  : 

22.  29  by 

41                24. 

36 

by  8i. 

26. 

33 

by  7|. 

23.  32  by 

31                25. 

91 

by  51 

27. 

54 

by6f 

28.  Divide  24  by  5f 

^4\   24 

^^    "  Explanation. — By  changing  the  dividend  and 

•  •  divisor  to  7ths,  the  division  is  performed  as  in  simple 

39  )168(4,^     ^^^^^^^^^- 
Divide  the  following : 

29.  36  by  7  f.  32.  59  by  4  |.  35.  39  by    7|. 

30.  46  by  3  |.  33.  36  by  3  |.  36.  35  by  16f 

31.  52  by  6^.  34.  96by43-V  37.  46  by    8f. 

38.  When  cloth  is  $.10|  per  yard,  how  many  yards  can 
be  bought  for  $3.36? 

39.  My  lot  has  a  frontage  on  the  street  of  385  feet.  How 
many  rods  is  it  in  front,  161  feet  being  a  rod  ? 

40.  A  cheap  kind  of  flour  sells  for  $3f  per  barrel.  How 
many  barrels  can  be  bought  for  %  76i  ?   . 

41.  A  farmer  bought  a  wagon  for  %  68,  paying  for  it  in 
apples  at  $1 J  per  barrel.  How  many  barrels  did  he  give 
for  it  ? 

42.  Boots  are  commonly  sold  in  cases  containing  12 
pairs.  If  a  dealer  paid  $  99  for  a  quantity  of  boots  at  %  2J 
a  pair,  how  many  cases  did  he  purchase  ? 


174  ELEMET^TS   OF   ARITHMETIC. 

186.   To  divide  a  fractiou  by  a  fraction. 

1 .  At  $  -|-  eachj  how  many  books  can  be  bought  for  $  |  ? 
At^l? 

2.  Among  how  many  children  can  -|  of  a  melon  be 
divided,  if  each  child  is  given  i  ?  |  ?  f  ? 

3.  If  cream  is  worth  $^  per  pint,  how  many  pints  can 
be  bought  for  $^?     If  $  j\  per  pint  ?     If  $  j\  ? 

4.  How  many  pitchers,  each  holding  ^j  of  a  gallon,  can 
be  filled  from  if  of  a  gallon  ? 

5.  If  a  man  uses  -^^  of  a  pound  of  butter  each  day,  how 
many  days  will  ^|  of  a  pound  last  him  ? 

6.  How  many  times  is  -f^  contained  in  i|-?  y4_  jn  -^-^? 
_3_  in  -9-  ?  -3_  in  la  (or  l")*?  -«-  in  1  ^  -2-  in  1  ^  -^  in  1  ^  -L 

1  2    ^^^    1  2   •      1  2    -"^V    1  2     \^^    ^)  ■      12^^^-^*      1  2    ^^^    -•-  •      1  2    -"^^    -•-  •      12 

in  -9-  *?  -2_  in  -6-  *?  -3-  in  -«-  ^ 

^^^12-      1  2    ^^M  2    •      1  2    -^^^    1  2    • 

7.  If  a  train  moves  |  of  a  mile  each  minute,  how  many 
minutes  will  it  require  to  go  3^  miles  ?  How  many  thirds 
are  there  in  3|-  miles  ? 

8.  If  it  takes  f  of  a  yard  of  lace  to  trim  a  bonnet,  how 
many  bonnets  can  be  trimmed  with  2|-  yards  of  lace  ? 

9.  How  many  aprons  can  be  made  from  4i  yards  of  mus- 
lin, if  each  apron  requires  1|-  yards  ? 

? 


How  many  half  yards  are  there  in  li  yards  ? 
How  many  half  yards  are  there  in  3^  yards 


10.  How  many  times  are  3  half  yards  contained  in  6 
half  yards  ? 

11.  How  many  books  can  be  bought  with  $6|,  at  $1| 
apiece  ? 

12.  How  many  pitchers,  each  holding  2 J  pints,  will  be 
required  to  hold  13|  pints  of  milk  ? 

13.  Among  how  many  children  can  i  of  a  pound  of  candy 
be  divided,  if  each  child  is  given  i-  of  a  pound  ? 

14.  How  many  eighths  are  there  in  ^  ? 

15.  If  a  coat  requires  |  of  a  dozen  buttons,  how  many 
coats  will  require  4^  dozen  buttons  ? 


FRACTIONS.  17e5 

16.  With  $f,  how  many  yaids  of  cloth  can  be  bought  at 
^f  per  yard? 

WRITTEN    EXERCISESo 

187.  1.  Divide  |-  by  |,  or  find  how  many  times  f  is  con- 
tained in  |-. 

7:_j_2._7.N^3_  2.1^  Explanation.  —  Since  it  is  required  to  find 

how  many  times  f  is  contained  in  |,  we  first 

find  how  many  times  |  is  contained  in  1,  and  then  find  |  of  that  result. 

Since  i  is  contained  in  1  3  times,  |  is  contained  in  1  one  half  of  3 
times,  or  |  times. 

Since  |-  is  contained  in  1  |  times,  in  |  it  will  be  contained  |  of  f 
times,  or  f  ^  times. 

188.  From  the  explanation  given,  it  is  apparent  that  : 

A  fraction  may  he  divided  by  a  fract'ion  by  7nultii:)lying  the 
dividend  by  the  divisor  inverted. 

1.  Use  cancellation  whenever  it  is  possible  to  do  so. 

2.  Reduce  mixed  numbers  and  integers  to  improper  fractions. 

2.   Divide  |  by  ^. 

Or,  f  =  fi,  and  f  =  f f .     There- 


fore, 


21 


189.   Hence  it  is  seen  that : 

Another  method  of  dividing  a  fraction  by  a  fractioyi  is  to 
reduce  the  fractions  to  similar  fractions,  and  then  to  divide  the 
numerator  of  the  dividend  by  the  numerator  of  the  divisor. 

Divide : 


3. 

fby,-V 

9. 

H  by  f 

15. 

26i  by  4f . 

4. 

Hby|. 

10. 

33-Vbyli 

16. 

19^  by  3f-. 

5. 

fbyf 

11. 

9A  by  2|. 

17. 

361  by  ^. 

6. 

Aby^. 

12. 

7A  by  41 

18. 

l^  by  f . 

7. 

A  by  f 

13. 

^  by  f 

19. 

16f  by  3f . 

8. 

Hby^V 

14. 

12|  by  7^. 

20. 

"\  by  If. 

17G  ELEMENTS   OF   ARITHMETIC. 

21.  How  many  times  may  2}r  gallons  be  drawn  from  a 
barrel  that  holds  31|^  gallons  ? 

22.  How  many  bushels  of  potatoes,  at  $ii  a  bushel,  may 
be  bought  with  $  5  ? 

23.  How  many  barrels  of  apples,  each  holding  2|  bushels, 
can  be  filled  with  18  bushels  ? 

24.  At  $  1|-  a  basket  for  peaches,  how  many  baskets  of 
peaches  shall  I  get  for  $  8  ? 

25.  When  flour  is  $  7|  per  barrel,  how  many  barrels  of 
ilour  can  be  bought  for  $  59  ? 

26.  How  many  pounds  of  beef,  at  11|/  per  pound,  can  be 
bought  for  871^? 

27.  How  many  books,  at  $31-  per  volume,  can  be  pur- 
chased for  $31^? 

28.  When  oranges  are  121^  per  dozen,  how  many  dozen 
can  be  bought  for  98f ^  ? 

29.  How  many  plows,  at  ^ll^-  each,  can  be  bought  for 
$67^? 

30.  How  many  acres  of  land,  at  $43|-  per  acre,  can  be 
bought  for  $237^? 

31.  A  railroad  16^  miles  long  cost  $66,937.     What  was 
the  cost  per  mile  ? 

32.  If  4i  hogsheads  hold  25|-  bushels,  what  will  1  hogs- 
head hold  ? 

33.  If  a  field  containing  541  acres  yielded  1165  bushels 
of  rye,  what  was  the  average  crop  per  acre  ? 

34.  Divide  |  of  f  of  f  of  ^  by  |  of  |  of  |. 

3''4''7''9J  •  U     3''8j-^^^''y''9''^''2''^"9 


Explanation.  —  In  solving  examples  like  this,  aU  the  factors  of  the 
divisor  should  be  inverted.  All  the  integers  and  mixed  numbers  should 
be  changed  to  improper  fractions  before  the  factors  of  the  divisor  are 
inverted.     Use  cancellation  to  abridge  the  process. 


FRACTIONS.  177 

Divide: 

36.  i  of  i  Of  I  of  tV  by  I  of  -I  of  ,\  of  I  of  6. 

36.  I  of  f  of  f  of  T^  by  |  of  f  of  f  of  8. 

37.  I  of  j\  of  t\  of  T^  by  i*  of  f  of  Jj  of  24. 

38.  Jjj  of  4  of  If  of  if  by  y\  of  i  of  \l  of  36. 

39.  I  of  f  of  iV  of  A  by  A  of  H  of  I  of  T*3  of  30. 

40.  tV  of  f  of  A  of  A  by  f  of  ^a  of  ^^  of  ^  of  44. 

41.  I  of  f  of  JJ-  of  I-  by  i  of  J_  of  if  of  ^  of  25. 

42.  I  of  t  of  i|  of  .*-,  by  I  of  T*5  of  ff  of  I  of  16. 

43.  f  of  JV  of  H  of  if-  by  4  of  J^  of  if  of  A  of  10. 

44.  T%  of  f  of  l-l  of  I  by  f  of  4  of  If  of  f  |  of  20. 
46.  f  of  f  of  if  of  /y  by  I  of  il  of  H  of  ^  of  45. 
46.  -rV  of  I  of  If  of  f  by  I  of  if  of  ^  of  If  of  18. 

190.  Expressions  of  unexecuted  division  of  fractions  are 
frequently  written  in  the  form  of  fractions.  Such  expres- 
sions are  sometimes  called  Complex  FractionSe 

3  02 

Thus,  f  ->  f  may  be  written  | ;  6f -^f,  -y-. 

7  9 

Ic    Find  the  value  of  the  fractional  form,  ^. 

IT 

4 

Sot  tttton  ^    —  1  _:_  _3_  _  44  ^  2.1  _  44  _  l  l  7 

TT 

Find  the  value  of  the  following : 


2. 

1 
1' 

5. 

5i 
4' 

8. 

2i 

11. 

n 

H 

3. 

1 

i 

6. 

9. 

If 

12. 

-|of  8 

4. 

EL. 

7 

OF    AR. 

5 

r 
—12 

10. 

5| 
3i' 

13. 

2t\ 
iofi 

178  ELEMENTS   OF   ARITHMETIC, 

191.   To  find  the  relation  of  one  number  to  another. 

1.  There  were  5  oranges  on  a  plate.  Willie  took  1 
orange,  and  Eddie  took  2.  What  part  of  all  the  oranges 
did  Willie  take  ?     Eddie  ? 

2.  What  part  of  8^  is  1^  ?     3^  ?     5^  ? 

7  is  -J-Q  of  10,  because  it  is  7  of  the  10  equal  parts  into 
which  10  may  be  divided.     What  part  of  10  is  9  ? 

3.  What  part  of  10  acres  are  5  acres  ?  Of  9  quarts  are 
3  quarts  ?  Of  8  pounds  are  2  pounds  ?  Of  10  inches  are  2 
inches  ? 

4.  1  is  what  part  of  12  ?    2  of  12  ?     3  of  12  ?     8  of  12  ? 

5.  5  melons  are  what  part  of  7  melons  "^ 

6.  $  5  are  what  part  of  $  8  ? 

7.  2  fifths  are  what  part  of  3  fifths  ? 

8.  3  eighths  are  what  part  of  7  eighths? 

9.  f  is  what  part  of  f  ?     ^-^  of  -f^  ?     ^\  of  \\  ? 

10.  What  part  of  2  ounces  is  f  of  an  ounce  ? 
Suggestion.  —  2  ounces  =  f  ounces.    What  part  of  |  is  J? 

11.  What  part  of  $  5  are  $  21  ? 

12.  What  part  of  \  of  a  gallon  is  i  of  a  gallon  ? 
Suggestion.  — i  =  f ;  i  =  f .     What  part  of  f  is  |  ? 


13.  What  part  of  10  quarts  are  3|-  quarts 

14.  What  part  of  7  acres  are  If  acres  ? 


15.  What  part  of  |-  of  a  rod  is  J  of  a  rod  ? 

16.  What  part  of  2|  pounds  are  li  pounds  ? 

17.  What  part  of  10  is  \  ?     Of  10  is  |  ?     Of  10  is  1\  ? 

WRITTEN    EXERCISES. 

192.    1.    What  part  of  $  100  are  $  72  ? 

^2   18  Explanation.  —  $1   is  j^-^  of  $  100  ;   consequently, 

TOO  —  25-      $ 72  are  f ^^^  of  $  100,  or  if  of  $  100. 

1.  When  one  or  both  numbers  are  fractions  not  having  a  common 
denominator,  the  numbers  should  be  reduced  to  equivalent  fractions 
having  a  common  denominator. 

2.  Mixed  numbers  should  be  reduced  to  improper  fractions. 


FRACTIONS.  179 

What  part  of : 

2.  85  is  30?  7.    25is2i?  12.      H  is  I  ^ 

3.  175  is  28?  8.    40is3|?  13.      -^is^? 

4.  12isf?  9.64is4i?  14.    SiVisli? 

5.  20  13  4?  10.    33isl|-?  15.      7iis4i? 

6.  32is|-?  11.    72is2f?  16.      5J  is  3|  ? 

17.  Ill  a  farm  of  120  acres  there  were  35J  acres  of  pasture 
land.     What  part  of  the  farm  was  used  for  pasture  ? 

18.  What  part  of  272^  square  feet  are  9  square  feet  ? 

19.  If  a  boy  can  do  a  piece  of  work  in  25  days,  what  part 
of  the  work  can  he  do  in  7f  days  ? 

20.  Miss  Reed  paid  $20|  for  a  gown,  and  $  61-  for  a  hat. 
What  part  of  the  price  of  the  dress  did  the  hat  cost  ? 

21.  If  the  AYorking  day  is  10  hours  long,  how  much  will 
a  man  who  earns  $  3  a  day  get  for  6  hours'  work  ? 

ALIQUOT  PARTS. 

193.   An  aliquot  part  of  a  number  is  any  integer  or  mixed 
number  that  will  exactly  divide  it. 

ALIQUOT  AND  OTHER  PARTS  OF  A  DOLLAR. 


50^  = 

25^  = 
20^  = 


16|^  =  fi. 


io^  =  f  iV 

66|^  =  f|. 

Hf  =  $^. 

75)«  =  $|. 

6if=9^. 

40/  =  «f 

5^=$i^. 

831/ =  $|. 

ill  =  ^j\. 

37i/  =  «|. 

2^  =  «!^V 

871/ =  fi. 

194.    1.    AYhat  will  12  pounds  of  butter  cost  at  33^^  per 
pound  ? 

Solution,  —  S3lf  =  $  }.    Since  1  pound  costs  $  i,  12  pounds  will  cost 
12  times  fi,- which  is  $  i/-,  or  $4. 


180  ELEMENTS   OF   ARITHMETIC. 

2.  What  will  20  readers  cost  at  25^  apiece  ?     At  50^  ? 

3.  What  will  25  sponges  cost  at  20^  apiece  ? 

4.  What  will  36  yards  of  ribbon  cost  at  16|^  a  yard  ? 

5.  What  will  48  yards  of  lace  cost  at  121)^  a  yard  ? 

What  is  the  cost  of  ; 

6.  120  pounds  of  sugar  at  5^  ?     At  SJ^  ?     At  W  ? 

7.  12  neckties  at  75p  ?     At  66fj^  ?     At  83^^  ? 

8.  40  baskets  at  37^^  ?     At  40^  ?     At  30^  ? 

9.  80  gallons  of  molasses  at  37J-^  ?   At  62^^  ?   At  87^^  ? 

10.  28  dozen  pencils  at  12^^  ?     At  16|^  ?     At  20^  ? 

11.  16  pairs  of  gloves  at  66|^  ?     At  83^^  ?     At  75^  ? 

12.  24  days'  work  at  1 1.25  ?     At  ^  1.331  ?     At  $  1.16|  ? 

13.  At  16|^  per  pound,  how  many  pounds  of  raisins  can 
be  bought  with  $  2  ? 

Solution.  —  16|/  =  $  |.  Since  1  pound  costs  1 1,  as  many  pounds 
can  be  bought  with  $  2  as  $  |  is  contained  times  in  $  2,  or  $  -^2,  which 
is  12  times.     Therefore,  12  pounds  of  raisins  can  be  bought  with  $2. 

14.  At  33^/  a  dozen,  how  many  dozen  eggs  can  I  buy 
with  $5?     $3?     $7?     $4? 

15.  At  37^)^  apiece,  how  many  books  can  be  bought  for 
$21?     $3|?     $41?     $51? 

16.  For  $  21,  how  many  yards  of  muslin  can  be  bought 
at  10^  a  yard  ?     At  12i^  ?     At  16|^  ? 

195.  To  find  the  whole  when  a  part  and  the  relation  of  the 
part  to  the  whole  are  given. 

1 .  If  $  5  is  ^  of  my  money,  how  much  money  have  I  ? 
10  cents  is  i  of  my  brother's  money.     How  much  has  he  ? 

2>  A  man  sold  5  gallons  of  vinegar,  which  was  i  of  all 
he  had.     How  many  gallons  had  he  ? 

3.  A  farmer  sold  10  quarts  of  milk  per  day,  which  was 
^  of  what  the  cows  produced.  How  much  did  they  pro- 
duce per  day  ? 


FRACTIONS,  181 

4.  A  boy  paid  $  6  for  a  coah,  and  that  sum  was  J  of  his 
wages  for  a  montli.     What  weie  his  monthly  wages  ? 

5.  A  lad  purchased  a  box  of  cartridges,  fired  off  10  of 
them,  and  had  -f^  of  them  left.  How  many  cartridges  were 
there  in  the  box  ? 

6.  A  boy  borrowed  50  feet  of  kite  string,  which  was  J-g 
of  what  another  boy  had.  How  many  feet  of  string  did 
the  other  boy  have  ? 

7.  A  farmer  estimated  that  a  hail  storm  had  caused  him 
a  loss  of  5  bushels  of  oats  per  acre,  or  yL  of  his  crop.  How 
much  did  he  think  his  land  Tvould  yield  him  per  acre  ? 

8.  A  boy  and  his  sister  played  backgammon  for  some 
time,  his  sister  beating  him  15  games,  which  was  \  of  the 
number  of  games  played.     How  many  did  they  play  ? 

9.  If  2  thirds  of  a  number  is  6,  hoAV  much  is  i  of  it? 

10.  If  3  fourths  of  a  number  is  9,  how  much  is  i  of  it? 

11.  If  5  sevenths  of  a  number  is  10,  how  much  is  -^  of  it? 
Since  ^  of  the  number  is  2,  what  is  the  number  ? 

12.  If  I  of  a  number  is  8,  how  much  is  ^  of  the  number  ? 
Since  ^  of  the  number  is  4,  what  is  the  number  ? 

WRITTEN    EXERCISES. 

196.    lo    320  is  I  of  Avhat  number  ? 

^  =  320.  Explanation.  —  Since  320  is  f  of  a  certain 

1 n4  number,  i  of  320,  or  64,  is  i  of  the  number;  and 

^            *  since  64  is  l  of  the  number,  the  number  must  be 

Number  =  512.  §  times  64,  or  512. 

Find  the  number  of  which 

2.  40  is    |.  8.  125  is  j%.  14. 

3.  60  is    f.  9.  340  is  ^.  15. 

4.  85  is  y%.  10.  560  is  f|.  16. 

5.  75  is    |.  llo  300  is  ^.  17. 

6.  6o  is    f  12.  240  is  i|.  18. 

7.  81  is  yV  13.  330  is  if  19. 


fis 

f 

^iris 

*• 

II  is 

1- 

If  is 

*• 

f«is 

f. 

tfis 

A- 

182  ELEMENTS   OF   ARITHMETIC. 

20.  A  boy  sold  his  bicycle  at  second  hand  for  |  of  the 
cost,  receiving  for  it  $  80.     How  much  did  it  cost  him  ? 

21.  Our  tennis  court  is  just  f  of  what  it  was  originally, 
and  it  is  now  120  feet.     How  long  was  it  originally  ? 

22.  The  distance  between  Kochester  and  Syracuse,  N.Y., 
is  81  miles,  and  it  lacks  4  miles  of  being  -J-  of  the  distance 
from  New  York  to  Buffalo.  What  is  the  distance  from 
New  York  to  Buffalo? 

23.  The  width  of  a  river  near  its  source  was  45  feet,  and 
that  distance  was  -^  of  what  it  was  20  miles  farther  down. 
How  wide  was  it  at  the  latter  point  ? 

24.  A  gardener  set  out  some  cabbage  plants,  but  only  80 
of  them  lived,  which  was  just  -^  of  the  number  set  out. 
How  many  did  he  set  out  ? 

25.  A  way  train  stopping  at  all  stations  averaged  25 
miles  per  hour ;  but  the  rate  of  the  train  was  only  ^  of  that 
of  a  fast  express  train.  How  many  miles  per  hour  did  the 
express  train  run  ? 

REVIEW    EXERCISES. 
ORAL    EXERCISES. 
197.   1.  Change    to   improper  fractions:     3|;     5|-;     6|; 

2.  Change  to  mixed  numbers:    -^;   ^-^-',   4=-^-;   ^;   -^; 

4  2  .     5J. 
6     5       9    • 

3.  Find  the  sum  of  3|,  4f,  51  6i  7f,  and  3f . 

4.  Find  the  sum  of  61  41,  2|,  3f,  41  and  2|. 

5.  A  man  had  3  lots  containing  respectively  3i  acres, 
2J  acres,  and  3|-  acres.  How  many  acres  did  they  all 
contain  ? 

6.  The  expense  of  a  carpet  was  $12|,  and  it  cost  ^31 
to  put  it  down.  What  was  the  entire  cost  of  the  carpet 
laid  in  the  room  ? 


FRACTIONS.  183 

7.  Terrence's  uncle  gave  him  $  10  on  Cliristmas,  but  he 
spent  $  of  of  it.     How  much  had  he  left  ? 

8.  A  father  allowed  his  son  $40  per  year  for  spending 
money.  At  the  end  of  the  year  the  boy  had  $  18|-.  How 
much  did  he  spend  ? 

9.  The  expense  of  a  Columbia  bicycle  was  $  140  ;  but  a 
boy  bought  one  that  pleased  him  just  as  well  for  -|  of  that 
price.     How  much  did  it  cost  him  ? 

10.  The  price  of  California  peaches  was  $  |  per  dozen  in 
July.     How  much  did  8  dozen  cost? 

11.  A  man  had  a  field  containing  4|^  acres,  which  he 
divided  into  lots  containing  -J  of  an  acre.  How  many  lots 
were  there  ? 

12.  A  w^agon  w^iich  cost  $  50^  was  sold  for  $  46J.  How 
much  was  the  loss  ? 

13.  After  purchasing  a  silver  watch  for  f  19f ,  I  had 
$  21|  left.     How  much  had  I  at  first  ? 

14.  A  man  owed  for  groceries  $18|-;  for  meat,  $  lOf , 
and  for  clothes  $S^.  If  he  paid  the  debts  out  of  $50, 
which  he  has,  how  much  did  he  have  left  ? 

15.  A  boy  went  on  a  visit  to  his  cousins  with  $  35  to  pay 
his  expenses.  On  returning  he  found  that  he  had  spent  -f- 
of  his  mone}^     How  much  had  he  left  ? 

16.  The  proprietor  of  a  boarding  house,  after  buying  pro- 
visions for  the  day,  found  that  he  had  $  351  left,  which  was 
just  -1-  of  the  money  he  had  before  making  the  purchases. 
How  much  did  he  spend? 

17.  An  American  flag  was  purchased  by  some  school 
children  for  $llf,  and  the  flag-staff  cost  $61  What  did 
both  cost  ? 

18.  A  man  sold  a  horse  for  $  160  and  a  cow  for  |  as  much. 
How  much  did  he  get  for  both  ? 

19.  A  cow  was  sold  for  ^  60,  which  was  |  of  what  a  horse 
was  sold  for.     How  much  was  received  for  both  ? 


184  ELEMENTS   OF    ARITHMETIC. 

20.  After  sj^ending  J  of  my  money,  I  had  $35  left. 
How  much  had  I  ? 

21.  A  horse  traveled  38^  miles  in  7  hours.  What  was 
his  average  rate  of  speed  per  hour  ? 

22.  A  man  sold  a  cow  for  $40  and  spent  -J-  of  the  sum. 
Afterwards  he  bought  a  horse  and  paid  in  cash  what  he  had 
left  from  the  sale  of  the  cow.  What  did  the  horse  cost 
him,  if  the  cash  payment  was  ^  of  the  cost  of  the  horse  ? 

23.  A  farm  was  sold  in  2  parts.  The  first  part  contained 
50  acres  and  that  was  -f-  of  the  second  part.  How  many 
acres  were  there  in  the  farm  ? 

24.  I  spent  i  of  my  money  for  books,  i  of  it  for  furniture, 
and  had  f  100  left.     How  much  had  I  at  first  ? 

25.  James  had  a  sum  of  money  such  that  i  of  it  was  just 
^  of  Henry's.  What  part  of  Henry's  was  the  whole  of 
James's  ? 

26.  One  half  of  a  mother's  age  was  f  of  her  daughter's. 
What  part  of  the  mother's  age  was  the  daughter's  ? 

27.  What  part  of  the  daughter's  age  was  the  mother's? 

28.  I  multiplied  by  another  fraction  gives  a  product  of  f. 
What  is  the  multiplier  ? 

29.  The  difference  between  i  and  J  of  a  number  is  8. 
What  is  the  number  ? 

WRITTEN    EXERCISES. 
198.   Find  the  value  of : 

1.  3i  +  34  +  G|-3|-A_5,^. 

2.  5i  +  6i  +  3TV-6f-5A  +  2.V 

3.  7f  +  5|-93L-|-8i-ii  +  5^. 

4.  (7-|  +  3|)x5i4-(3H-U)x3i 

5.  (3|-li)-l|  +  (2i-li)x6^-5. 

6.  (4f  +  6}  -  3f )  ~  A  +  (31  -  21)  X  6. 

7.  (4i-3i+7TV  +  f +  4)-J^  +  8. 

8.  (0f  +  4|)x(5i-3i)-(2i-lf). 


FRACTIONS.  .  185 

9.    How  much  will  5  masons  earn  in  4 J-  clays,  at  $  3|  per 
day? 

10.  After  selling  i  of  liis  load  of  potatoes  to  one  person, 
i  to  another,  and  25  bushels  to  another,  a  farmer  had  sold 
his  entire  load.     How  many  bushels  did  he  sell  ? 

11.  One  fifth  of  the  expenses  of  an  excursion  were  paid 
by  Mr,  A.,  i  of  them  by  Mr.  B.,  and  the  rest,  which  was 
$  17,  by  the  rest  of  the  party.  What  were  the  expenses  of 
the  excursion  ? 

12.  A  man  forgot  how  many  bushels  of  rye  he  had  sold, 
but  he  remembered  that  he  had  received  ^  185  for  the 
rye  at  ^  J  per  bushel.     How  many  bushels  did  he  sell  ? 

13.  A  man  loaned  some  money,  receiving  at  the  end  of 
the  year  $  245  interest  for  it.  If  the  rate  of  interest  was 
j-f  0  of  the  sum  loaned,  how  much  did  he  loan  ? 

14.  A  man  finds  that  he  must  save  ^600  per  year.  He 
takes  boarders,  and  can  accommodate  10.  Suppose  that  he 
pays  $  2520  for  provisions,  servants,  and  rent,  and  his  rooms 
are  full  all  the  year ;  how  much  must  he  charge  per  week 
to  save  $  600  ? 

15.  A  man  of  means  spent  -|  of  his  life  traveling  in 
America,  \  of  it  in  Europe,  and  the  rest  of  it  at  his  own 
home.  AVhat  part  of  his  life  did  he  spend  at  his  own 
home  ? 

16.  If  he  spent  30  years  at  his  own  home,  how  old  was 
he? 

17.  If  A  can  do  a  piece  of  work  in  6  da3^s,  what  part  of 
it  can  he  do  in  1  day  ?  If  B  can  do  the  same  piece  of 
work  in  7  days,  what  part  of  it  can  he  do  in  1  day  ?  If 
botli  work  together,  what  part  of  it  can  they  do  in  1  day  ? 

18.  If  both,  working  together,  could  only  do  -^  of  it  in  1 
day,  how  many  days  would  they  need  to  complete  it? 
Since  both  together  can  do  Jf  in  1  day,  what  part  of  42 
days  will  they  need  to  complete  it  ? 


186  ELEMENTS   OF   ARITHMETIC. 

19.  A  man  bequeathed  J  of  his  property  to  his  brother, 
■1-  of  it  to  his  sister,  and  the  rest,  which  was  $  62500,  to  a 
college.     How  much  did  he  bequeath  in  all  ? 

20.  A  can  do  a  piece  of  work  in  3  days,  and  B  can  do  it 
in  5  days.     In  what  time  can  both  together  do  it  ? 

21.  A  can  do  a  piece  of  work  in  8  days,  and  B  can  do  it 
in  10  days.     In  what  time  can  both  do  it  together  ? 

22.  It  takes  A  and  B  8  days  to  do  a  piece  of  work  which 
B  can  do  alone  in  12  days.     In  what  time  can  A  do  it  ? 

23.  Two  men  during  a  year  earned  together  $1085;  but 
one  of  them  only  earned  f  as  much  as  the  other.  How 
much  did  each  earn  ? 

24.  Two  brothers  earn  respectively  $2  and  $3  per  day. 
In  what  time  can  both  together  earn  $  13  ? 

25.  A  man  who  owned  f  of  a  sash  factory  sold  -|  of  his 
share  for  $  24000.  How  much  was  the  factory  worth  at 
that  rate  ? 

26.  Divide  |  of  f  of  i|  of  41  by  f  of  |  of  ^j  of  if 

27.  Divide  f  of  f  of  |  of  6f  by  ^  of  |  of  f  of  2f. 

28.  Find  the  sum  and  the  difference  of  4iand  3f ;  divide 
the  sum  by  the  difference ;  the  difference  by  the  sum ;  and 
then  find  the  product  of  the  quotients. 

29.  Multiply  3f  by  2|;  divide  3f  by  2|;  and  find  the 
difference  between  the  two  results. 

30.  A  man  spent  1-  of  his  money  and  $25  more,  and  then 
had  $  225  left.     How  much  money  had  he  ? 

31.  A  man  after  spending  $2250  in  repairing  his  house, 
found  that  it  cost  him,  including  repairs,  just  J  of  the  origi- 
nal cost.     What  was  the  original  cost  ? 

32.  The  estimated  cost  of  a  building  was  J  of  its  actual 
cost.  If  J  of  the  estimated  expense  was  $  5000,  Avhat  was 
the  actual  cost  of  the  building  ? 

33.  A  school  in  one  of  our  cities  had  the  names  of  1039 
pupils  on  its  rolls.     If  there  had  been  one  pupil  more,  there 


FRACTIONS.  /      A^^ 

would  have  been  2\  times  as  many  girls  aJ^luo^lfin  the 
school.    How  many  boys  and  how  many  girls  were  enrolled  ? 

34.  A  man  engaged  in  manufacturing  made  a  profit  of 
yl^Q  on  his  capital,  and  this  sum,  added  to  his  capital,  made 
i  27060.  How  much  capital  had  he  invested  in  the  busi- 
ness ? 

35.  The  product  of  three  numbers  is  72|.  Two  of  them 
are  5f  and  9|.     What  is  the  other  number  ? 

36.  If  a  man  walks  S^-  miles  in  2f  hours,  how  far  can  he 
walk  in  5|-  hours  ? 

37.  A  meter,  the  French  unit  of  length,  is  very  nearly 
39f  inches.  What  is  the  difference  in  length  between  1 
yard  and  1  meter  ?  Wliat  is  the  difference  between  5^ 
yards  and  5|-  meters  ? 

38.  My  annual  income  from  money  loaned  at  $  6  on 
f  100  is  I  36.60.     How  much  have  I  loaned  ? 

39.  A  company  owned  a  cotton  factory  worth  $  350,000. 
If  A  and  B  own  -f-  of  it,  what  is  the  value  of  their  share  ? 

40.  Find  the  difference  between  401  and  25|;  divide  the 
difference  by  the  sum  and  determine  the  quotient. 

41.  If  6  is  added  to  both  terms  of  the  fraction  |,  is  its 
value  increased  or  diminished,  and  how  much  ? 

42.  A  merchant  sold  i  i  and  ^  of  his  tea  and  had  130 
pounds  left.     How  much  tea  had  he  ? 

43.  The  cost  of  a  live  turkey  weighing  14  pounds  was 
$  2.  If  the  waste  in  dressing  was  f  of  the  weight  of  the 
turkey,  for  how  much  per  pound  must  I  sell  it  to  gain  $  1 
on  the  cost  ? 

44.  One  half  of  my  goods  was  destroyed  by  fire ;  i  the 
remainder  was  destroyed  by  water  and  the  rest  cost  me 
$  8650.     What  was  the  entire  cost  of  the  goods  ? 

45.  A  carpenter  alone  can  build  a  house  in  25  days,  but 
with  the  assistance  of  his  son  he  can  build  it  in  15  days. 
In  how  many  days  can  the  son  build  it  ? 


DECIMAL  FRACTIONS. 


199.  1.  If  anything  is  divided  into  10  equal  parts,  what 
is  each  part  called  ? 

2.  If  -f^  of  a  line  is  divided  into  10  equal  parts,  what  part 
of  the  whole  line  is  each  23art  ?     How  much  is  -^  of  -f^? 

3.  If  y^-g-  of  a  line  is  divided  into  10  equal  parts,  what 
part  of  the  line  is  each  part  ?     How  much  is  yL  of  y^^-  ? 

4.  How  many  hundredths  are  equal  to  J^-? 

5.  How  many  thousandths  are  equal  to  yi^-  ? 

6.  The  divisions  of  anything  into  tenths,  hundredths, 
thousandths,  etc.,  are  called  decimal  divisions. 

200.  One  or  more  of  the  decimal  divisions  of  a  unit  is 

called  a  Decimal  Fraction. 

The  word  dea'mal  is  derived  from  the  Latin  word  decern,  wliich 
means  ten. 

Decimal  fractions  are  usually  called  decimals. 

201.  Since  decimals  have  the  same  law  of  increase  and 
decrease  as  integers,  the  denonmiator  of  the  fraction  may 
be  indicated  by  the  x>osition  of  the  figures. 

202.  The  figures  in  the  first  place  at  the  right  of  units 
represent  tenths;  in  the  second,  hundredths ;  in  the  third, 
thousandths ;  in  the  foiirth,  ten-thousandths  ;  etc. 

203.  A  period  called  the  Decimal  Point  is  i)laced  before 
the  decimal. 

Thus,  .9  represents  -^^       .48  represents  -^^^       .059  represents  j||o 
.2  represents  ^^       .08  represents  yf^       .007  represents  j-^^^ 

188 


DECIMAL   FRACTIONS. 


189 


.5  represents  j%       .04  represents  jf  ^  .006  represents  yoW 

.8  represents  j\     1.09  represents  lyf  ^  .029  represents  yfli^ 

1.1  represents  ly^  l-^^  represents  ly^^  5.005  represents  Sy^^^ff 

Numeration  Table. 


<o 

.c 

^ 

to       c 

-=    s 

c 

+J           CO 

..:      -o        3 

dreds. 

s. 

:s. 

"1 

"5 
.5 

c/5 

dredths. 
usandths 
•thousan 
dred-tho 

j= 

c      c     .t: 

"S 

"c 

c        o        c        c 

3          OJ          c 

s^ 

<1> 

Z3         J=           <D           = 

3=      (—       =3 

Q 

H- 

31         1-         1-         = 

s 

7  9  6 

5 

8  4  6  9 

4 

INTEGERS. 

DECIMALS. 

The  orders  below  millionths  in  their  order  are  : 
dred-millionths,  billionths,  ten-billionths,  hundred 

204.   What  order  of  decimals  occupies 
1st  place  ?  2d  place  ? 

3d  place  ?  4tli  place  ? 

5tli  place  ?  6th  place  ? 

What  decimal  place  is  occupied  by 
Tenths  ?  Ten-thousandths  ? 

Thousandths  ?       Millionths  ? 
Hundredths  ?        Hundred-thousandths  ? 


ten-millionths,  hun- 
-billionths,  etc. 


7th  place  ? 
2d  place  ? 
4th  place  ? 

Ten-millionths  ? 
Hundredths  ? 
Thousandths  ? 


EXERCISES   IN    NUMERATION. 

205.  1.  Eead  42.356. 

Explanation.  —  The  decimal  part  of  the  number  expresses  3  tenths, 
5  hundredths,  6  thousandths,  or  356  thousandths.  The  whole  expres- 
sion is,  therefore,  read  :  42  and  356  thousandths. 

206.  Hence  it  is  evident  that : 

III  reading  a  decimal,  the  decimal  slioidd  be  read  as  an  integer, 
and  the  denomination  of  the  rigJit-hand  figure  should  he  added. 

In  reading  an  integer  and  decimal,  use  the  word  and  i»nly  between 
the  integral  and  decimal  parts  of  the  number. 


190  ELEMENTS   OF   ARITHMETIC. 

Eead  the  following : 


2. 

.31- 

9. 

2.63 

16. 

217.305 

3. 

.73 

10. 

3.10 

17. 

301.1031 

4. 

.281 

11. 

5.031 

18. 

130.2005 

5. 

.032 

12. 

7.003 

19. 

800.80061 

6. 

.0578 

13. 

72.070  ■ 

20. 

256.0032 

7. 

.5601 

14. 

26.303 

21. 

250.00071 

8. 

.5013 

15. 

31.120 

22. 

504.03062 

EXERCISES  IN    NOTATION. 

207.  1.    Express  decimally  54  thousandths. 

Explanation.  —  Since  54  thousandths  is  equal  to  5  hundredths  and 
4  thousandths,  4  is  written  in  the  thousandths'  place,  5  in  hundredths' 
place,  and  since  there  are  no  tenths,  0  in  tenths'  place.  A  decimal 
point  is  then  placed  before  the  tenths.     Hence  54  thousandths  =  .054. 

208.  Hence  it  is  evident  that : 

In  expressing  a  fraction  decimally,  the  numerator  of  the  frac- 
tion is  written  and  ciphers  are  prefixed,  if  necessary,  to  indicate 
the  denominator.    The  decimal  point  is  then  placed  before  tenths. 

Express  the  following  as  decimals : 

2.  Six  tenths.     Four  tenths.     Nine  tenths. 

3.  Thirteen  hundredths.     Thirty  hundredths. 

4.  Fifteen  thousandths.     Two  hundred  one  thousandths. 

5.  Seventeen  thousandths.     Six  ten-thousandths. 

6.  Fourteen    and    three    hundred    thirty-two   hundred- 
thousandths.      Five  millionths. 

7.  Fifty-one  and  seven  hundred  forty-two  millionths. 

8.  Fifty  and  seventy -five  ten-thousandths. 

9.  Thirty-one  and  sixty-four  hundred-thousandths. 

10.  Seventy-nine  and  nine  ten-millionths. 

11.  Three  hundred  and  three  millionths. 

12.  Sixty  and  sixty-seven  millionths. 

13.  312  millionths.     201  ten-millionths. 


DECIMAL   FRACTIONS.  191 

14.  705  ten-millionths.     605  and  703  billiontlis. 

15.  631  ten-billioiitlis.     25  and  25  ten-millionths. 

16.  ^Vo-  19-  AVo-  22.  23710^^^. 
17-  jiih.  20.  210^11^^.  23.  6751^^^. 
18.  6^^.               21.    601^1-1^.           24.    3002^VoW 


REDUCTION  OF  DECIMALS. 

209.  To  reduce  dissimilar  decimals  to  similar  decimals. 

1.  How  do  the  fractions  y%,  -f^-^,  and  yWo"  compare  in 
value  ?  How  do  the  decimals  expressing  them  compare  in 
form  ?  What  is  the  effect  of  annexing  a  cipher  to  a 
decimal  ? 

2.  How  do  the  fractions  -^-q,  j^-q,  and  y-oVo  compare  in 
value  ?  How  do  the  decimals  expressing  them  compare  in 
form  ?  What  is  the  effect  of  prefixing  a  cipher  to  a  decimal  ? 

3.  How  does  the  number  of  places  in  a  decimal  compare 
with  the  number  of  ciphers  in  the  denominator  ? 

210.  Hence  it  is  evident  that: 

1.  Annexing  ciphers  to  a  decimal  does  not  cdter  its  value. 

2.  Each  decimal  cipher  prefixed  to  a  decimcd  diminishes 
the  value  of  the  decimal  tenfold. 

3.  The  number  of  places  in  a  decimcd  is  equcd  to  the  number 
of  ciphers  in  the  denominator  of  its  equivcdent  coinmon  fraction. 

WRITTEN    EXERCISES. 

211.  1.    Eeduce  .6,  .46,  and  .095  to  similar  fractions. 

Explanation.  —  Since  the    lowest    order  of  the 

PAA      given  decimals  is  thousandths,  all  the  decimals  must 

=  .DUU      i^g  changed  to  thousandths.     Since  annexing  ciphers 

46    =  .460      to  a  decimal  does  not  alter  its  value,  we  give  to  each 

^Q95  __  Q95       number  three   decimal   places   by  annexing  ciphers, 

thus  making  them  similar  by  reducing  them  all   to 

thousandths. 


192  ELEMENTS   OF   ARITHMEXrC. 

2.  Reduce  to  similar  decimals :  .0321,  .00G23,  .00025,  .75. 

3.  Eeduce  to  ten-thousandths:  .68,  .0785,  .063,  .251,  .27. 

4.  Express  8  tenths  as  hundredths  ;  as  thousandths. 

5.  Change  to  similar  decimals  :  .003,  .02,  .2100,  .6530. 

212.  To  reduce  a  decimal  to  a  common  fraction. 

WRITTEN    EXERCISES. 
1.    Eeduce  .75  to  its  equivalent  common  fraction. 

Explanation.  —  .75  expressed  as  a  common  frac- 
.75  =  yVo"  ^^  I     *^^^  ^®  tW'  which,  being  reduced   to   its  smallest 
terms,  is  |. 

213.  From  the  explanation  it  is  apparent  that : 

A  decimal  may  he  reduced  to  a  common  fractioyi  by  omitting 
the  decimal  point  and  siqyj^lying  the  proper  denominator. 

The  result  should  be  expressed  in  its  lowest  terms. 

Reduce  the  following  to  common  fractions  in  their  smallest 
terms  : 


2. 

.75. 

6. 

.125. 

10. 

.0045. 

14. 

.00375. 

3. 

.50. 

7. 

.875. 

11. 

.075. 

15. 

.00625. 

4. 

.35. 

8. 

.625. 

12. 

.00875. 

16. 

.05375. 

5. 

.80. 

9. 

.375. 

13. 

.0250. 

17. 

.0875. 

214.  To  reduce  a  common  fraction  to  a  decimal. 

1.  How  many  tenths  are  there  in  i?     In  i? 

2.  How  many  hundredths  are  there  in  \  ?    In  i  ?    In  J? 

3.  How  many  hundredths  are  there  in  1  ?     In  ^  ?     In  |-  ? 

4.  How  many  thousandths  are  there  in  1  ?     In  i?     In  |-? 

WRITTEN   EXERCISES. 

215.  1.    Reduce  |-  to  an  equivalent  decimal. 

ft\  ^  C\C\(\  Explanation.  —  |  is  \  of  5,  or  of  50  tenths ;  and  \  of 

1 — ^ 50  tenths  is  6  tenths,  and  2  tenths  remaining.    2  tenths  are 

.625      equal  to  20  hundredths,  and  i  of  20  hundredths  is  2  hun- 
dredths and  4  hundredths  remaining.     4  hundredths  are 
equal  to  40  thousandths,  and  i  of  40  thousandths  is  5  thousandths. 
Hence  |  is  equal  to  G  tenths  +  2  hundredths  -1-  5  thousandths,  or  .625. 


DECIMAL   FRACTIONS.  193 

216.   From  the  process  it  is  evident  that : 

A  common  fraction  may  be  reduced  to  a  decimal  by  annex- 
ing ciphers  to  the  numerator  and  dividing  by  the  denominator, 
and  pointing  off  as  many  decimal  places  in  the  quotient  as 
there  are  ciphers  annexed. 

In  many  cases  the  division  is  not  exact.  In  such  instances  the 
remainder  may  be  expressed  as  a  common  fraction,  or  tlie  sign  +  may 
be  employed  after  the  decimal  to  show  that  the  result  is  not  complete  ; 
thus  1  =  .166 1,  or  .166  +  . 

Reduce  the  following  to  decimals : 


2. 

3 

To"' 

6. 

A- 

10. 

i*. 

14. 

f 

18. 

If 

22. 

f 

3. 

5 

8- 

7. 

A- 

11. 

¥k- 

15. 

1  5 
4T- 

19. 

8|. 

23. 

5 
T2- 

4. 

tV 

8. 

4V 

12. 

3^ 
1  1- 

16. 

^15- 

20. 

^, 

24. 

A- 

5. 

A. 

9. 

A. 

13. 

A. 

17. 

3i. 

21. 

6A. 

25. 

"sf-o- 

ADDITION  OF  DECIMALS. 


217.    lo    What  is  the  sum  of  ^^  and  ^-^  ?     Of  .4  and  .7  ? 

2.  What  is  the  sum  of  j^  and  -^\  ?     Of  .08  and  .09  ? 

3.  What  is  the  sum  of  ^L  and  j^^?     Of  .5  and  .25 ? 


WRITTEN   EXERCISES. 
218.    1.  'What  is  the  sum  of  2.85,  1.625,  and  3.0235  ? 

9  S^AA        9  Q^  Explanation. — The  numbers  are  written  so 

Z.boOO  or  ^-oo^  ^^^^  ^j^i^g  ^^^^-^^  under  units,  tenths  under  tenths, 
1.6250  1.625  etc.  The  decimals  may  be  made  similar  by  annex- 
3.0235       3.0235    i^g  ciphers  and  then  added,  care  being  taken  to 

—    place  the  decimal  point  in  the  sum  directly  under 

7.4985        7.4985    the  decimal  point   in  the  numbers  added.     Or, 
since  only  units  of  one  order  are  found  in  any 
column,  it  is  not  necessary,  in  practice,  to  make  the  decimals  similar. 

Find  the  sum  of  the  following : 

2.  8.75  +  .369  +  75.212.  4.  3.801  +  2.875  +  1-625. 

3.  1.89  +  62.625  +  3.939.         5.  1.25  +  4.275  +  48.35. 

EL.    OF    AR.  —  13 


194  ELEMENTS   OF  ARITHMETIC. 

6.  .356  + 1.275  +  32.0325.  11.  3.495  +  1-87  +  3.25. 

7.  15.815  +  17.21  +  31.41.  12.  38.835  +  8.9764  +  1.385„ 

8.  .2735  +  1.596  +  3.25.  13.  1.8035  +  3.00412  +  31.025, 

9.  2.948  +  3.7642  +  85.035.  14.  62.589  +  3.875  +  1.3759. 
10.  46.2  +  381.25  +  2.985.  15.  28.48  +  17.578  +  34.752. 

16.  What  is  the  sura  of  8  tenths,  12  hundredths,  75  thou- 
sandths, 69  ten-thousandths,  21  hundredths  ? 

17.  What  is  the  sum  of  312  ten-thousandths,  8  mil- 
lionths,  65  thousandths,  413  ten-millionths,  9  tentlis  ? 

18.  What  is  the  sum  of  82  hundredths,  510  thousandths, 
6  tenths,  80  ten-thousandths  ? 

19.  What  is  the  sum  of  303  thousandths,  4108  millionths, 
635  ten-thousandths,  803  ten-millionths  ? 

20.  A  bicyclist  traveled  on  his  wheel  51.875  miles  in 
one  day,  45.75  the  next,  and  35.375  the  next  day.  How  far 
did  he  travel  during  these  three  days  ? 

21.  A  ticket  agent  sold  in  one  month  tickets  to  the  value 
of  $  18,756.75,  in  the  next  $  17,983.25,  in  another  $  2001.875. 
How  much  were  his  sales  during  those  three  months  ? 

SUBTRACTION  OP  DECIMALS. 

219.  1.    From  j\  take  -\.     From  .18  take  .13. 

2.  From  ji-^  take  j^^.     From  .035  take  .026. 

3.  From  4  take  -f\.     From  8  take  .8. 

WRITTEN  EXERCISES. 

220.  1.    From  46.57  subtract  9.46325. 

46  ^7000  —  4rC  ^7  Explanation.  —  The  numbers  are  written 

^  ~       *  so  that  units  stand  under  units,  tenths  under 

y.46o2o  =    9.463-.5      tenths,   etc.     The  decimals  may   be   made 

oT^A/^r^-'       o^^^TaT^^      similar  and  then  subtracted,  care  beinff  taken 
oT.lObiO      o<.10o<5     ^1       .1     J     •      1       •  ^  '    ^v-  -J 

to  place  the  decimal  ponit  m  the  remainder 

directly  under  the  decimal  point  in  the  num- 
ber subtracted.  The  ciphers  may  be  supposed  to  be  annexed  when 
we  subtract,  and  consequently  need  not  be  written. 


DECIMAL   FRACTIONS,  196 


2. 

3. 

4. 

5. 

6. 

From 

25.38 

39.85 

87.35 

46.81 

24.37 

Take 

14.05 

29.755 

59.355 

21.35 

12.358 

7.  8.  9.  10.  11. 

From     72.189         78.895         26.825         34.24  21.12 

Take      35.976         53.5987       21.3452       30.16  12.31 

12.  What  is  the  difference  between  6.325  and  1.0345  ? 

13.  What  is  the  difference  between  6.45  and  2.3375  ? 

14.  From  65  hundredths  subtract  65  thousandths. 

15.  From  162  thousandths  subtract  261  millionths. 

16.  Find  the  value  of  23.2  -  .0325  +  6.95  -  3.4876. 

17.  Find  the  value  of  6.594  -  2.0341  -  .70325  +  8.963. 

18.  Find  the  value  of  81.8  +  35.625  -  38.875  -  2.0034.    , 

19.  Find  the  value  of  4.6  +  2.635  -  .8925  -  .0359. 

20.  From  a  cistern  that  contained  30.5  barrels  of  water, 
25.75  barrels  were  drawn  off.  How  much  water  remains  in 
the  cistern  ? 

21.  A  man  bought  a  tub  of  butter  for  $35.75,  paying 
$  10  in  cash  and  the  rest  iii  groceries.  How  much  did  he 
pay  in  groceries  ? 

22.  A  man  whose  income  was  $  10,000  spent  in  one  year 
$  7375.87.     How  much  did  he  save  that  year  ? 

MULTIPLICATION    OP    DECIMALS. 

221.  1.  AYhat  is  the  product  of  ^\  x  yV^  C)f  .1  X  .1  ? 
Of  .5  X  .5  ?  How  does  the  number  of  ciphers  in  the  de- 
nominator of  the  product  of  these  fractions  compare  with 
the  number  of  ciphers  in  the  denominators  of  the  factors  ? 
2.  What  is  the  product  of  j\  Xj^^?  Of  .1  x  .01? 
Of  .5  x  .05  ?  How  does  the  number  of  ciphers  in  the  de- 
nominator of  the  product  of  these  fractions  compare  with 
the  number  of  ciphers  in  the  denominators  of  the  factors  ? 


196  ELEMENTS  OF   ARITHMETIC. 

3  How  does  the  number  of  decimal  places  in  a  decimal 
fraction  compare  with  the  number  of  ciphers  in  its  denomi- 
nator ? 

4.  How,  then,  does  the  number  of  decimal  places  in  the 
product  of  two  or  more  decimals  compare  with  the  number 
of  places  in  the  factors  ? 

222.    It  is  apparent,  therefore,  that : 

The  number  of  decimal  places  in  the  product  of  tico  or 
more  decimals  is  equcd  to  the  number  of  decimal  places  in  the 
factors. 

WRITTEN   EXERCISES. 

,     223.    1.   What  is  the  product  of  .275  multiplied  by  .17? 

.275         Explanation.  — Since  275  and  17  are  the  numerators  of 

^rj     tlie  fractions,  in  mipltiplying  tlie  fractions,  we  must  multiply 

''         275  by  17,  which  gives  4675  for  a  new  numerator. 

192o         Since    thousandths    multiplied    by   hundredths    produce 

275       hundred-thousandths,  the  product  of  the  fractions  is  4675 

~C)AC7K     hundred-thousandths,  or  .04675. 

Or  the  number  of  decimal  places  in  the  product  will  be 
five,  the  number  of  decimal  places  in  both  factors.     Art.  222. 

224.   Hence,  it  is  evident  that : 

To  multiply  decimcds  tee  midtipjly  as  if  the  numbers  were 
integers,  and  from  the  right  of  the  product  point  off  as  many 
figures  for  decimcds  as  there  are  decimal  places  in  both 
factors. 

If  the  product  does  not  contain  a  sufficient  number  of  decimal  places, 
the  deficiency  must  be  supplied  by  prefixing  ciphers. 

Multiply : 

2.  .25  by  .75.  5.  .75  by  2.06.  8.  .876  by  .375. 

3.  .24  by  3.4.  6.  .93  by  .047.  9.  .433  by  2.75. 

4.  .57  by  68o  7.  .39  by  .875.  10.  72.2  by  .055. 


DECIMAL   FRACTIONS.  197 

11.  3.62  by  .0037.  15.  2.37  by  7.563.       19.  .1895  by  .0436. 

12.  1.98  by  2.034.  16.  2.53  by  .00635.     20.  6.81  by  0.0875. 

13.  25.7  by  3.887.  17.  .3147  by  .0052.     21.  578  by  .0934. 

14.  15.8  by  .0855.  18.  .1963  by  .00638.  22.  0.765  by  .067. 

23.    Multiply  5.372  by  100. 

Explanation.  —  Since  each  removal  of  a  figure  one 

5.372  place  to  the  left  increases  its  value  tenfold,  the  removal  of 

100      the  decimal  point  one  place  to  the  right  multiplies  by  10, 

537.200      and  its  removal  two  places  to  the  right  multiplies  by  100  ; 

or,  the' process  may  be  performed  in  the  ordinary  manner. 

24.  Multiply  27.65  by  10.  28.  Multiply  6.8735  by  100. 

25.  Multiply  49.675  by  10.  29.  Multiply  .3768  by  1000. 

26.  Multiply  2.795  by  100.  30.  Multiply  6.835  by  100. 

27.  Multiply  8.935  by  100„  31.  Multiply  .7634  by  1000. 

32.  Multiply  89.92  by  10.    By  100.    By  1000.    By  10,000. 

33.  Multiply  68.3025  by  100.     By  1000.     By  10,000. 

34.  Which  is  larger  and  how  much,  the  product  of  .625 
and  .375,  or  their  sum  ? 

35.  What  will  52.75  bushels  of  oats  cost  at  ^.45  per 
bushel  ? 

36.  A  merchant  sold  24.25  yards  of  muslin  at  ^.125  a 
yard.     How  much  did  he  receive  for  it  ? 

37.  A  man  bought  214  bags  of  barley.  What  did  they 
cost  him,  if  each  bag  held  2.5  bushels,  and  the  price  was 
$  .96  a  bushel  ? 

38.  A  tailor  bought  4.5  yards  of  cassimere  at  $2.25  a 
yard,  and  6.25  yards  of  broadcloth  at  $  6.50  a  yard.  How 
much  more  did  the  broadcloth  cost  than  the  cassimere  ? 

39.  How  many  acres  are  left  in  a  farm  of  563.27  acres, 
if  3  fields  of  57.135  acres  each  are  sold  from  it  ? 

40.  A  steamship  in  crossing  the  Atlantic  sailed  at  an 
average  speed  of  325.75  miles  per  day.  If  another  steamer 
sailed  from  the  same  port  at  the  same  time  at  the  rate  of 
395.35  miles  per  day,  how  far  were  they  apart  in  5.75  days? 


198  ELEMENTS   OF   ARITHMETIC. 


DIVISION  OF  DECIMALS. 

1.    When  a  number  is  divided  by  3,  what  part  of  it 
is  found  ?     By  6?     By  8  ?     By  15  ? 

2.  How  much  is  i  of  9  ?     9--3?     JoflS?     15^5? 

3.  How  much  is  i  of  9  tenths  ?  i  of  .8  ?  i  of  .6  ?  i  of  .6  ? 

4.  How  much  is  i  of  25  hundredths  ?  i  of  .24  ?  i'^of  .36  ? 

WRITTEN   EXERCISES. 
226.    1.    Find  1  sixth  of  109.92,  or  divide' 109.92  by  6. 
6)109.92  Explanation,  —  One  sixth  of  10  tens  is  1  ten  and  4 

— ^  o  09  tens  remaining  ;  1  sixth  of  4  tens  remaining  and  9  units, 
or  49  units,  is  8  units  and  1  unit  remaining  ;  1  sixth  of  1 
unit  remaining  and  9  tentlis,  or  19  tenths,  is  3  tenths  and  1  tenth 
remaining ;  1  sixth  of  1  tenth  remaining  and  2  hundredths,  or  12  hun- 
dredtlis,  is  2  hundredtlis.  Therefore,  one  sixtli  of  109.92  is  18.32,  or 
the  quotient  of  109.92  .-^  6  is  18.32. 

2.  What  is  1  fifth  of  86.4055  ?  5.  $  234.54  --  9  =  ? 

3.  What  is  1  eighth  of  94076.8  ?        6.  $  907.34  --  7  =  ? 

4.  What  is  1  eleventh  of  1086.415?  7.  $324,784-8  =  ? 
8.  Find  1  eighth  of  9675  acres. 

8)9675.000         Explanation,  —  We   divide   as  in  simple   division 
1209  375     ^^^^^^  ^®  come  to  the   remainder,  3  acres.     This  we 
change  to  30  tenths  and  continue  the  division  thus: 
1  eighth  of  30  tentlis  is  3  tenths  and  6  tenths,  or  60  hundredths, 
remainder.     1  eighth  of  60  hundredths  is  7  hundredths  and  4  hun- 
dredths, or  40  thousandths,  remainder  ;  1  eighth  of  40  thousandths  is  5 
thousandths.     Therefore,  1  eighth  of  9675  acres  is  1209.375  acres. 

How  much  is : 

9.    i  of  $23.50?  12.  i  of  392.25?  15.  8.643^6? 

10.  i  of  $36.75?  13.  i  of  468.36?  16.  85.86 -- 9  ? 

11.  i  of  $59.64?  14.  -1- of  693.34?  17.  3.845 -- 5  ? 

18.  A  man  received  $3278  for  5  acres  of  hand.     What 
was  the  price  per  acre  ? 

19.  An  estate  vahied  at  $35,267  was    divided  equally 
among  8  children.     What  was  each  child's  share  ? 


DECIMAL   FRACTIONS.  199 

227o   The  number  of  decimal  places  in  the  quotient. 

1.  What  is  the  product  of  .5  x  .7  ?  How  many  decimal 
places  does  it  contain  ?  If  .35  is  divided  by  .7,  what  will  the 
quotient  be  ?     How  many  decimal  places  does  it  contain  ? 

2.  What  is  the  product  of  .9  x  .05  ?  If  .045,  the  product, 
is  divided  by  .9,  one  of  the  factors,  what  is  the  other  factor  ? 
How  many  decimal  places  does  it  contain  ? 

3.  Since  the  product  contains  as  many  decimal  places  as 
there  are  in  all  the  factors,  if  the  product  of  two  factors  is 
given  and  one  of  the  factors,  how  may  the  number  of  decimal 
places  in  the  other  factor  be  found  ? 

4.  Since  the  dividend  may  be  regarded  as  the  product  of 
two  factors,  one  of  whicli  is  the  divisor,  and  the  other  the 
quotient,  how  may  the  number  of  decimal  places  in  the 
quotient  be  found  from  those  in  the  dividend  and  divisor  ? 

228.  It  is  therefore  evident  that : 

The  quotient  tcill  contain  as  many  decimal  places  as  the 
number  of  decimal  ]^laces  in  the  dividend  exceeds  the  number 
in  the  divisor. 

WRITTEN   EXERCISES. 

229.  1.    Divide  .15652  by  .043. 

.043). 15652 (3.64         Explanation. — Since  the  dividend  is  equal 

-j^29  to  the  product  of  the   divisor  and  quotient,  if 

-  „     ■  15652,  the  numerator  of  the  dividend,  is  divided 

by  43,  the  numerator  of  the  divisor,  the  result 

-^58  obtained  will  be  the  numerator  of  the  quotient. 

172  Dividing,  it  is  found  to  be  SG-l.     Hence,  the 

-^^2  numerator  of  the  quotient  is  found  precisely  as 

in  simple  division.     The  denominator  of  the-' 

quotient  may  be  found  by  dividing  the  denominator  of  the  dividend  by 
the  denominator  of  the  divisor,  that  is  by  dividing  100000  by  1000, 
which  gives  100.     Therefore,  the  quotient  is  ff^,  or  3.64.     Or, 

The  quotient  will  contain  as  many  decimal  places  as  the  number  of 
decimal  places  in  the  dividend  exceeds  the  number  in  the  divisor. 
Art.  228. 


200 


ELEMENTS   OF  ARITHMETIC. 


230.   Hence  it  is  evident  that : 

In  dividing  decimals,  ive  divide  as  if  the  numhers  ivere 
integer's,  and  from  the  right  of  the  quotient  point  off  as  many 
places  for  decimcds  as  the  number  of  decimcU  places  in  the 
dividend  exceeds  the  number  in  the  divisor. 

1.  If  the  quotient  does  not  contain  a  sufficient  number  of  decimal 
places,  the  deficiency  must  be  supplied  hj  prefixing  ciphers. 

2.  Before  commencing  the  division,  the  number  of  decimal  places 
in  the  dividend  slioidd  he  made  at  least  equal  to  the  number  of  decimal 
places  in  the  divisor. 

3.  When  there  is  a  remainder  after  using  all  the  figures  of  the 
dividend,  annex  decimal  ciphers  and  continue  the  division. 

4.  For  business  purposes,  it  is  not  necessary  to  carry  the  division 
further  than  to  four  or  five  decimal  figures  in  the  quotient. 


Divide : 

2.  78.12  by  3.6. 

3.  12.32  by  3.12. 

4.  1675.8  by  .49. 

5.  11.904  by  3.1 

6.  452.92  by  .052. 

7.  63.168  by  .94. 

8.  .15275  by  .325. 

9.  .12376  by  .0364. 

10.  .0184428  by  .5123. 

11.  .29184  by  .0057. 
12„  4.5872  by  .0122. 

13.  .056448  by  15.68. 

14.  .0816785  by  6.235. 

15.  .0983253  by  .0673. 

16.  5.97066  by  35.73. 

32.  Divide  325.47  by  100. 


17.  14.91  by  70, 

18.  8.25  by  1.5. 

19.  .00864  by  .24. 

20.  107.5  by  .043. 

21.  .0009  by  .125, 

22.  .07245  by  .23. 

23.  800  by  .0004. 

24.  12.126  by  .235. 

25.  167.544  by  71.6. 

26.  56.112  by  0.56. 

27.  2.1828  by  .034. 

28.  4066.2  by  .648. 

29.  .08  by  1.611. 

30.  114.87  by  .0035. 

31.  30000  by  .00003. 


100)325.47         Explanation.  —Since  each  removal  of  a  figure  one 

"3.2547 
divides  by  10,  and  the  removal  two  places  by  100 


o  9r  ir-     place    to    the   right  decreases  its   vahie   tenfold,   the 
""^  removal  of  the  decimal  point  one  place  to  the  left 


DECIMAL  FRACTIONS. 


201 


231o    Hence  it  is  evident  that ; 

To  divide  by  1  ivith  any  numher  of  ciphers  annexed,  the 
decimal  point  should  he  removed  as  many  places  to  the  left  as 
there  are  ciphers  annexed  to  the  1. 


33.  Divide  392.5  by  100. 

34.  Divide  26.45  by  10. 

35.  Divide  369.5  by  1000. 


36.  Divide  4.825  by  100. 

37.  Divide  38.62  by  1000. 

38.  Divide  42.85  by  10000„ 


39.  How  many  yards  of  clotli,  at  $  5.75  per  yard,  can  be 
bought  for  ^  153.525  ? 

40.  How  many  barrels  of  apples,  at  $  1.25  per  barrel,  can 
be  bought  for  $  i8.43|  ? 

41.  How  many  cords  of  wood,  at  $  5. 62 J  per  cord,  can  be 
bought  for  $1495.121? 


ACCOUNTS  AND  BILLS. 

232.  A  record  of  the  debts  and  credits  between  two 
parties  is  called  an  Account. 

233.  A  statement  of  the  quantity  and  price  of  the  arti- 
cles sold,  and  the  value  of  the  whole,  is  called  a  Bill. 

A  bill  is  receipted  when  the  words  Received  Payment,  or 
Paid,  are  written  at  the  bottom,  and  the  seller's  name  is 
signed  either  by  himself  or  by  some  authorized  person. 

234.  The  following  abbreviations  are  in  common  use : 


@, 

At. 

Do., 

The  same. 

Mdse., 

Merchandise 

%, 

Account. 

Doz., 

Dozen. 

No., 

Number. 

Acc't, 

Account. 

Dr., 

Debtor. 

Pay't, 

Payment. 

Bal., 

Balance. 

Fr't, 

Freight. 

Pd., 

Paid. 

Bbl., 

Barrel. 

Hhd., 

Hogshead. 

Per, 

By. 

B(Vt, 

Bought. 

In.st. , 

This  month. 

Rec'd, 

Tfeceivedo 

Co., 

Company. 

Int., 

Interest. 

Yd., 

Yard. 

Cr., 

Creditor. 

Lb., 

Pound. 

Yr., 

Year. 

202  ELEMENTS   OF   ARITHMETIC. 

1.  BrooUyn,  J^.Y.,  June  25,  1892. 

Bought  of  JONES    <Sc    PURDY. 


4-  {>^ou(f^  <S.c^Pf&&, 

800  /^. 

@     f.Soh 

f  f06 

60 

d  Jvtatoy  S'&a^, 

260    '' 

.87\ 

227 

60 

8  {^-aoc&Qy  f^aui'i'nQ^, 

200    ^' 

-          J2\ 

i        26 

00 

2  6~6^t.  (^afj&&-Q.iic^cLV, 

^32   '' 

.06\ 

27 

00 

2  6~6Z.  ^la^k&w^, 

/■50    ''        "         ,06h 

8 

26 

R&(^&i 

\/d9^ 

26 

Make  ont  in  proper  form,  find  the  footings  of  the  follow- 
ing bills  and  accounts,  and  receipt  them  : 

2.  Miss  Lucy  Graham  bought  of  Luckey,  Piatt  &  Co.  15 
yards  of  calico  @  10  cents;  36  yards  of  sheeting  @  18^ 
cents;  2  pairs  of  gloves  @  $1.50;  1  sun-umbrella  @  %  3.75; 
5  yards  of  Hamburg  edging  @  25  cents ;  6  handkerchiefs  @ 
37i  cents ;  and  7  pairs  of  hose  @  50  cents. 

3.  Messrs.  Henry  Davis  &  Co.  bought  of  James  Harkness 
150  barrels  of  flour  @  %  5.90;  105  bushels  of  wheat  @  f  1.20 ; 
325  bushels  of  corn  @  $.68;  675  bushels  of  oats  @  |  .40; 
and  50  barrels  of  potatoes  @  %  1.25. 

4.  Mrs.  Charles  Reid  in  account  with  Roger  Caldwell: 
August  7, 1892,  9  yards  cashmere  @  $  .75  ;  \  yard  velvet  @ 
$1.50;  12  yards  lawn  @  .121;  li.  yards  silesia  @  $  .30;  | 
yards  silk  at  $  1.50. 

5.  Darwin  Howard  bought  of  Stone  &  Bacon  15  pounds 
nails  @  4ij^;  4  pairs  of  hinges  @  37|J^ ;  18  dozen  screws  @ 
11^;  5  dozen  milk-pans  @  $2.75;  7  locks  @  75^;  2  saws 
@  $  1.35 ;  3  hammers  @  '^^(f^ ;  and  2  planes  @  $  1.50. 


REVIEW.  203 

RBVIEWo 
WRITTEN   EXERCISES. 
235.    lo    How  much  will  2  dozen  pairs  of  kid  gloves  cost 
at  1 1.25  a  pair  ? 

2.  A  man  paid  out  the  following  sums :  For  a  pair  of 
horses,  $375;  for  a  carriage,  $295;  for  harnesses,  $115, 
How  much  did  he  pay  for  all  ? 

3.  A  farmer  sold  12.35  cords  of  wood  at  $  1.75  per  cord. 
How  much  did  he  receive  for  it  ? 

4.  A  man  purchased  8  tons  of  coal  at  $5.75  per  ton, 
and  7  cords  of  hard  wood  at  $5.50  per  cord.  How  much 
did  he  pay  for  fuel  ? 

5.  If  illuminating  gas  is  sold  at  $2.50  per  thousand 
cubic  feet,  for  how  much  gas  will  $  17.50  pay  ? 

6.  Find  the  sum  of  15  dollars  9  cents  3  mills,  12  dollars 
5  cents  4  mills,  18  dollars  14  cents  7  mills,  and  16  dollars 
10  cents. 

7.  If  5.5  yards  of  ribbon  cost  $2.75,  what  will  9.75 
yards  cost  ? 

8.  If  a  clerk  earns  $  520  per  year  and  spends  $  110  for 
clothes,  $215  for  board,  and  $  75.50  for  other  expenses,  how 
long  will  it  take  him  to  save  $  1000  ? 

9.  If  54,600  letters  were  mailed  from  a  post-office  in 
30  days,  what  was  the  average  number  mailed  per  day  ? 

10.  How  many  pairs  of  curtains  can  be  bought  for  $414, 
if  12  pairs  are  bought  for  $  138  ? 

11.  A  drover  bought  280  head  of  cattle  for  $  12,740,  and 
sold  them  at  a  gain  of  $8.75  per  head.  How  much  did  he 
receive  for  them  per  head  ? 

12.  A  man  paid  $87.50  for  materials  to  paint  his  house, 
and  he  paid  3  painters  $  2.50  per  day  for  4.5  days  to  do  the 
work.  How  much  was  the  entire  expense  of  painting  his 
house  ? 


204  ELEMENTS   OF  ARITHMETIC. 

13.  The  distance  between  two  places  is  168|  miles.  How 
long  will  it  take  a  bicyclist  to  ride  the  distance  at  12.5  miles 
per  hour  ? 

14.  How  much  will  one  dozen  each  of  cups  and  saucers 
cost,  if  the  cups  cost  $  .74J  and  the  saucers  $  .67-i-  apiece  ? 

15  „  What  must  I  pay  for  3  tables  at  $  12.75  each,  2  chairs 
at  $9.50  each,  6  chairs  at  $3.75  each,  and  one  sofa  at 
$25.50? 

16.  A  merchant  sold  35.5  yards  of  cambric  at  25  cents 
per  yard  and  gained  $  1.775.  How  much  did  it  cost  him 
per  yard  ? 

17 o  During  the  Civil  War  a  dollar  bill  was,  at  one  time, 
worth  only  65  cents  in  gold.  How  many  dollars  in  gold 
were  520  dollars  in  paper  money  worth  at  that  time  ? 

18.  A  farmer  received  for  his  barley  $151.37,  for  his 
oats  $  196.84,  for  his  corn  as  much  as  for  barley  and  oats, 
for  his  rye  $  75.93,  and  for  his  wheat  as  much  as  for  all  the 
others.     Eor  how  much  did  he  sell  his  grain  ? 

19.  A  dealer  bought  25  barrels  of  flour,  each  containing 
196  pounds,  for  $112.50,  How  much  did  it  cost  him  per 
pound  ?  How  much  will  be  his  entire  gain  if  he  sells  it  at 
3J  cents  per  pound  ? 

20.  A  merchant  paid  the  following  sums,  on  the  average, 
for  expenses  during  each  month  :  For  rent,  $  275 ;  for  heat- 
ing and  lighting,  $115;  for  clerk  hire,  $375;  and  for  other 
expenses,  $  S5.  What  will  be  his  net  gain  annually,  if  his 
gross  profits  are  $  12,560  ? 

21.  An  ice-dealer  put  into  his  ice-house  1315  tons  of  ice. 
He  sold  at  different  times  the  following  quantities :  216.75 
tons,  396.5  tons,  542.7  tons.  If  the  rest  melted  during  the 
summer,  how  many  tons  melted,  and  how  many  tons  did  he 
sell? 

22.  Ha  freight  train  runs  15.75  miles  per  hour,  in  how 
many  hours  will  it  run  189  miles  ? 


DENOxMINATE    NUMBERS. 


236.  A  number  used  in  connection  with  some  specified 
thing  is  called  a  Concrete  Number. 

Thus,  5  days,  6  men,  4  feet,  8  pounds  are  concrete  numbers. 

237.  A  number  used  without  reference  to  any  particular 
thing  is  called  an  Abstract  Number. 

Thus,  5,  6,  4,  8,  are  abstract  numbers. 

238.  A  concrete  number  in  which  the  unit  of  measure  is 
established  by  law  or  custom  is  called  a  Denominate  Number. 

Thus,  5  gallons,  8  rods,  5  acres,  are  denominate  numbers. 

239.  A  denominate  number  which  is  composed  of  units  of 
one  denomination  only  is  called  a  Simple  Denominate  Number. 

Thus,  3  feet,  8  ounces,  5  quarts,  are  simple  denominate  numbers. 

240.  A  denominate  number  which  is  composed  of  units 
of  two  or  more  denominations  that  are  related  to  each  other 
is  called  a  Compound  Denominate  Number. 

Thus,  5  yards  2  feet  8  inches  is  a  compound  denominate  number. 

MEASURES  OP  LENGTH. 

241.  In  measuring  length  or  distance,  Measures  of  Length 
or  Linear  Measures  are  used. 

Table. 
12  inches  (in.)  =  1  foot  (ft.). 
3  feet  =r  1  yard  (yd.). 

5J  yards!  ^^  rod  (rd.). 

16^  feet    /  ^      ^ 

320  rods  =  1  mile  (mi.). 

206 


206 


ELEMENTS  OF  ARITHMETIC. 


EXERCISES. 

1.  How  many  in.  are  there  in  2  ft.  ?    In  3  ft.?   In  4  ft.? 

2.  How  many  ft.  are  there  in  4  yd.  ?    In  5  yd.  ? 

3.  How  many  yd.  are  there  in  4  rd.  ?    In  6  rd.  ?    In  8  rd.? 

4.  How  many  ft.  are  there  in  2  rd.  ?   In  5  rd.  ?  In  10  rd.? 

5.  How  many  rd.  are  there  in  2  mi.  ?     In  3  mi.  ? 

6.  How  many  ft.  are  there  in  24  in.  ?    In  36  in.  ? 

7.  How  many  yd.  are  there  in  9  ft.  ?    In  15  ft.?    In  24  ft.? 

8.  How  many  rd.  are  there  in  33  ft.  ?     In  66  ft.  ? 

9.  How  many  mi.  are  there  in  640  rd.  ?     In  960  rd.  ? 

10.  How  many  in.  are  there  in  -J-  ft.  ?    In  |  ft.  ?    In  f  ft.  ? 

11.  How  many  ft.  are  there  in  i  yd.?     In  li  yd.  ? 

12.  How  many  in.  are  there  in  2  ft.  ?     In  2  ft.  6  in.  ? 

13.  How  many  ft.  are  there  in  1  yd.  ?    In  1  yd.  2  ft.  ?    In 
2  yd.  ?     In  2  yd.  1  ft.  ?     In  3  yd.  2  ft.  ? 

14.  How  many  in.  are  there  in  1  yd.?    In  1  yd.  1  ft.? 


Angle. 


MEASURES  OF  SURFACE. 

242.  Anything  that  has  only  length  and  breadth  is  called 
a  Surface. 

Thus,  this  page,  the  side  of  a  block,  etc.,  are  surfaces. 

243.  The  difference  in  the  direction 
of  two  lines  that  meet  is  called  an 
Angle. 

244.  A  figure  that  has  four  equal 
sides  and  four  equal  angles  is  called  a 
Square. 

1.  A  square  inch  is  a  square  each  of  whose 
sides  is  an  ijich  long  ;  a  square,  foot  is  a  square 
each  of  whose  sides  is  a/ooi  long. 

2.  The  angles  of  a  square  are  called  right 
angles. 


Square. 


DENOMINATE   NUMBERS. 


207 


Table. 

144  square  inches  (sq.  in. )  =  1 

ft      c^^no^.r^     f,^r^i■  1 


9  square  feet 
30|-  square  yards  \ 
272^  square  feet    / 
160  square  rods 
640  acres 


)  =  1  square  foot  (sq.  ft.), 
=  1  square  yard  (sq.  yd.)o 

=  1  square  rod  (sq.  rd.), 

=  1  acre  (A.). 

=  1  square  mile  (sq.  mi.)o 


EXERCISES. 

1.  How  many  sq.  in.  are  there  in  2  sq.  ft.  ?     In  3  sq.  ft.  ? 

2.  How  many  sq.  ft.  are  there  in  3  sq.  yd.  ?    In  4  sq.  yd.  ? 

3.  How  many  sq.  rd.  are  there  in  2  A.  ?     In  3  A.  ? 

4.  How  many  sq.  yd.  are  there  in  18  sq.  ft.  ?    In  36  sq.  ft.? 
6.    How  many  sq.  ft.  are  there  in  288  sq.  in.?  In 432  sq.  in.? 

6.  How  many  A.  are  there  in  J  sq.  mi.  ?     In  i  sq.  mi.  ? 

7.  How  many  sq.  rd.  are  there  in  ^  A.  ?     In  i  A.  ? 

8.  How  many  sq.  ft.  are  there  in  i  sq.  yd.  ?     In  |  sq.  yd.  ? 

9.  How  many  sq.  in.  are  there  in  ^  sq.  ft.  ?     In  i  sq.  ft.  ? 
10.  How  many  sq.  ft.  are  there  in  2  sq.  ft.  12  sq.  in.  ? 

In  5  sq.  ft.  80  sq.  in.  ? 


MEASURES  OP  VOLUME. 

245.  Anything  that  has  length,  breadth,  and  thickness 
is  called  a  Solid. 

246.  The  number  of  solid 
units  in  a  body  is  called  its 
volume  or  solid  contents. 

247.  A  solid  having  six  equal 
square  sides,  called  faces,  is  a 
Cube. 

A  sohd  whose  faces  are  each  a 
square  inch  is  a  cubic  inch;  a  solid 
whose  faces  are  each  a  square  foot  is 
a  cubic  foot. 


Cube. 


208  ELEMENTS  OF  ARITHMETIC. 

Table. 

1728  cubic  inclies  (cu.  in.)  =  1  cu.  ft  (cu.  ft.), 
27  cubic  feet  =  1  cubic  yard  (cu,  yd.), 

128  cubic  feet  =  1  cord  of  wood  or  stone  (C.)o 

EXERCISES. 

1.  How  many  cu.  in.  are  there  in  2  cu.  ft.  ?    In  5  cu.  ft.  ? 

2.  How  many  cu.  ft.  are  there  in  4  cu.  yd.  ?     InScu.  yd.  ? 

3.  How  many  cu.  ft.  are  there  in  2  C.  ?     In  3  C.  ? 

4.  How  many  cu.  yd.  are  there  in  54  cu.  ft.  ?   In  81  cu.  ft.  ? 

5.  How  many  cu.  in.  are  there  in  i  cu.  ft.  ?    In  i  cu.  ft.  ? 

6.  How  many  cu,  ft.  are  there  in  \  C?     In  |  C.  ? 

7.  How  many  cu.  ft.  are  there  in  2  cu.  yd.  3  cu.  ft.  ? 

8.  How  many  cu.  in.  are  there  in  1  cu.  ft.  72  cu.  in.  ? 

LIQUID   MEASURES. 

248.  Measures  used  in  measuring  all  fluids  and  liquids 
are  called  Liquid  Measures. 

Table. 

4  gills  (gi.)  =  1  pint  (pt.). 
2  pints  =  1  quart  (qt.). 

4  quarts       =  1  gallon  (gal.). 

1.  In  determining  the  capacity  of  cisterns,  reservoirs,  etc.,  31  i  gal- 
lons are  considered  a  barrel  (bbl.),  and  2  barrels,  or  63  gallons,  a 
hogshead  (hhd.). 

2.  A  gallon  contains  231  cubic  inches. 

EXERCISES. 

1 .  How  many  gi.  are  there  in  2  pt.  ?    In  3  pt.  ?    In  5  pt.? 

2.  How  many  pt.  are  there  in  3  qt.  ?    In  5  qt.  ?    In  8  qt.  ? 

3.  How  many  qt.  are  there  in  5  gal.  ?     In  6  gal.  ? 

4.  How  many  gal.  are  there  in  24  qt.  ?     In  40  qt.  ? 

5.  How  many  qt.  are  there  in  16  pt.  ?     In  20  pt.  ? 

6.  How  many  gal.  are  there  in  16  pt.  ?     In  20  pt.  ? 


DENOMINATE   NUMBERS.  209 

7  0  How  many  pt.  are  there  in  ^  gal.  ?     In  i  gal.  ? 

8.  How  many  pt.  are  there  in  3  qt.  ?     In  3  qt;  1  pt.  ? 

9.  How  many  gi.  are  there  in  1  pt.  3  gi.  ?    In  1  pt.  2  gi.  ? 
10.  How  many  pt.  are  there  in  1  gal.  ?     In  2  gal.  3  qt,  ? 

DRY  MEASURES. 

249.  Measures  used  in  measuring  fruit,  roots,  vegetables, 
and  grain  are  termed  Dry  Measures. 

Table. 

2  pints  (pt.)=l  (qt.). 

8  quarts        =  1  peck  (pk.), 

4  pecks  =  1  bushel  (bu. ) . 

1.  A  bushel  contains  2150.4  cubic  inches. 

2.  A  quart  and  a  pint  dry  measure  contain  a  little  more  than  the 
same  liquid  measures. 

EXERCISES„ 

1.  How  many  pt.  are  there  in  5  qt.  ?     In  7  qt.  ? 

2.  How  many  qt.  are  there  in  5  pk.  ?     In  4  pk.  ? 

3.  How  many  pk.  are  there  in  3  bu.  ?     In  6  bu.  ? 

4.  HoAv  many  bu.  are  there  in  24  pk.  ?     In  36  pk.  ? 

5.  How  many  pk.  are  there  in  40  qt.  ?     In  32  qt.  ? 

6.  How  many  qt.  are  there  in  16  pt.  ?     In  20  pt.  ? 

7.  How  many  pk.  are  there  in  i  bu.  ?     In  i  bu.  ? 

8.  How  many  pk.  are  there  in  2  bu.  3  pk.  ? 

AVOIRDUPOIS  WEIGHT. 

250.  In  weighing  groceries  and  all  heavy  articles  as  hay, 
grain,  coal,  including  all  metals  except  gold  and  silver,  Avoir- 
dupois Weight  is  used. 

Table, 
16  ounces  (oz.)  =  1  pound  (lb,). 

100  pounds  =  1  hundred-weight  (cwt.). 

20  hundred-weight  |  _  ^  ^qj^  /rp  ^ 
2000  pounds  i 

EL.  OF   AR.  —  14 


210  ELEMENTS   OF  ARITHMETICo 

EXERCISES. 

1.  How  many  oz.  are  there  in  2  lb.  ?    In  3  lb.  ?    In  4^  lb.  ? 

2.  How  many  lb.  are  there  in  4  cwt.  ?     In  6  cwt.  ? 

3.  How  many  lb.  are  there  in  3  T.  ?    In  5  T.  ?    In  61  T.  ? 

4.  How  many  cwt.  are  there  in  300  lb.  ?    In  400  lb.  ? 

5.  How  many  T.  are  there  in  2500  lb.  ?     In  3000  lb.  ? 
In  5000  lb.  ? 

6.  How  many  lb.  are  there  in  i  T.  ?     In  i  T.  ?     In  f  T.  ? 

7.  How  many  oz.  are  there  in  yL  T.  ?     In  y-J-Q  T.  ? 

TROY  WEIGHT. 

251.   In  weighing  gold,  silver,  and  jewels  Troy  Weight  is 

used. 

Table. 
24  grains  (gr.)     =  1  pennyweight  (pwt.). 
20  pennyweights  =  1  ounce  (oz.). 
12  ounces  =  1  pound  (lb.). 

The  Troy  lb.  contains  5760  gr. ;  the  Avoirdupois  lb. ,  7000  gr. 
The  Troy  oz.  contains  480  gr,  ;  the  Avoirdupois  oz,,  437^  gr, 

EXERCISES. 

1.  How  many  gr.  are  there  in  2  pwt,  ?     In  4  pwt.  ? 

2.  How  many  pwt.  are  there  in  2  oz.  ?    In  4  oz.  ?■  In  6  oz.  ? 

3.  How  many  oz.  are  there  in  4  lb.?    In  4i-  lb.?    In  51  lb.? 

4.  How  many  gr.  are  there  in  1  oz.  ?    In  ^  oz.  ?    In  i  oz.  ? 

5.  How  many  lb.  are  there  in  24  oz.  ?     In  36  oz.  ? 

6.  How  many  oz.  are  there  in  40  pwt.  ?     In  50  pwt.  ? 

7.  HoAv  many  pwt.  are  there  in  48  gr.  ?     In  60  gr.  ? 

8.  How  many  oz.  are  there  in  1  lb.  ?     In  2  lb.  4  oz.  ? 


DENOMINATE   NUMBERS.  211 


APOTHECARIES'  WEIGHT. 

252.  In  weighing  medicines  for  prescriptions  Apotheca- 
ries' Weight  is  used  by  apothecaries  and  physicians. 

Table. 

20  grains  (gr.)=  1  scruple  (sc,  or  3). 
3  scruples        =  1  dram  (dr.,  or  3)- 
8  drams  =  1  ounce  (oz.,  or  §). 

12  ounces  =  1  pound  (lb.,  cr  lb). 

EXERCISES. 

1.  How  many  gr.  are  there  in  3  sc.  ?    In  4  sc.  ?    In  5^  sc.  ? 

2o  How  many  sc.  are  there  in  4  dr.  ?    In  5  dr.  ?    In  10  dr.  ? 

3.  HoAV  many  dr.  are  there  in  3  oz.  ?    In  8  oz.  ?    In  9^  oz.  ? 

4o  How  many  oz.  are  there  in  2  lb.  ?    In  4i-  lb.  ?    In  5|  lb.  ? 

5.  How  many  lb.  are  there  in  48  oz.  ?  In  60  oz.  ?  In  66  oz.  ? 

6.  How  many  oz.  are  there  in  24  dr.  ?  In  40  dr.  ?  In  52  dr.  ? 

7.  How  many  dr.  are  there  in  12  sc?  In  18  sc.  ?  In  22  sc.  ? 

8.  How  many  sc.  are  there  in  40  gr.?  In50gr.  ?  In90gr.  ? 

9.  How  many  dr.  are  there  in  2  oz.  3  dr.  ?     In  3  oz.  5  dr.  ? 

DIVISIONS   OP   TIME, 

253.  Table. 

60  seconds  (sec.)  =  1  minute  (min.). 

60  minutes  =  1  hour  (hr.). 

24  hours  =  1  day  (da.). 

7  days  =  1  week  (wk,). 

365  days  —  1  year  (yr.). 

366  days  =  1  leap  year  (1.  yr.). 
100  years  =  1  century  (C). 

Centennial  years  whose  number  is  exactly  divisible  by  400,   and 
other  years  whose  number  is  exactly  divisible  by  4,  are  Leap  Years. 


212  ELEMENTS  OF   ARITHMETIC. 

There  are  12  months  in  a  year.     The  names  and  the  num- 
ber of  days  in  each  are  as  follows  : 

1.  January  (Jan.)  31  days.  7.  July  (July)  31  da. 

2.  February  (Feb.)  28  or  29  da.  8.  August  (Aug.)  31  da. 

3.  March  (Mar.)  31  da.  9.  September  (Sept.)  30  da. 

4.  April  (Apr.)  .30  da.  10.  October  (Oct.)  31  da. 

5.  May  (May)  31  da.  11.  November  (Nov.)  30  da. 

6.  June  (June)  30  da.  12.  December  (Dec.)  31  da. 

The  29th  day  of  February  is  the  day  added  to  make  the  leap  year. 

The  following  rhyme  will  serve  to  fix  in  memory  the  days 
in  each  month : 

Thirty  days  hath  September, 

April,  June,  and  November. 

All  the  rest  have  thirty-one, 

Save  February,  w^hich  alone 

Hath  twenty-eight,  and  one  day  more 

"We  add  to  it  one  year  in  four. 

EXERCISES. 

1.  How  many  sec.   are  there  in  5  min.  ?     In  6  min.? 
In  10  min.  ? 

2.  How   many   min.    are    there   in  2  hr.  ?      In  3  hr.  ? 

3.  How  many  hr.  are  there  in  2  da.  ?    In  4  da.  ?    In  i  da.  ? 

4.  How  many  da.  are  there  in  3  wk.  ?     In  6  wk.  ? 

5.  How  many  wk.  are  there  in  14  da.  ?     In  35  da.  ? 

6.  How  many  hr.  are  there  in  240  min.  ?     In  180  min.  ? 

7.  How  many  min.  are  there  in  120  sec.  ?     In  300  sec? 

8.  How  many  da.  are  there  in  4  wk.  3  da.  ?     In  5  wk.  ? 

9.  Which  of  the  following  named  years  are  leap  years? 
1880?  1892?  1900?  2000?  1864?  1870? 

10.    How  many  da.  are  there  in  June  ?     In  Dec.  ?     In 
Mar.  ?     In  Aug.  ?     In  Jan.  ?     In  Sept.  ?     In  Feb.  ? 


DENOMINATE   NUMBERS.  213 

COUNTING. 

254.  The  following  denominations  are  used  in  countingo 

Table. 
12  things  =  1  dozen  (doz.). 
12  dozen  =  1  gross  (gr. ) . 
12  gross   =  1  great  gross  (G.  gr.). 
20  things  =  1  score. 

STATIONERS'  TABLE. 

255.  The  following  table  is  used  in  the  paper  trade '. 

Table. 
24  sheets     =  1  quire  (qr.). 
20  quires     =  1  ream  (R.). 

2  reams     =  1  bundle. 

5  bundles  =  1  bale. 

EXERCISES. 

1.  How  many  things  are  there  in  3  doz.  ?     In  6  doz.  ? 

2.  How  many  are  3  score  ?     3  score  and  10  ? 

3.  How  many  doz.  are  there  in  3  gr.  ?     In  1  G.  gr.  ? 

4.  How  many  sheets  are  there  in  2  qr.  ?     In  5^  qr.  ? 

5.  How  many  qr.  are  there  in  3  K.  ?   In  6  R.  ?    Ine^R.? 

6.  How  many  qr.  are  there  in  48  sheets  ?    In  96  sheets? 


256. 

MISCELLANEOUS  EXERCISES. 

1.    4  bu. 

=  ?  pk. 

11. 

IJcu. 

ft. 

=  ?  cu.  in. 

2.    3  ft. 

=  ?  in. 

12. 

32  qt. 

=  ?  pk. 

3.    5  1b. 

=  ?  oz.  (Ap.) 

13. 

2iA. 

=  ?  sq.  rd. 

4.    10  qt. 

=  ?  pt. 

14. 

3  yd. 

=  ?  in. 

5.    18  sq. 

ft. 

=  ?  sq.  yd. 

15. 

48  qt. 

=  ?  gal. 

6.    6  oz. 

=  ?  pwt. 

16. 

31  hr. 

=  ?  min. 

7.    70  da. 

=  ?  wk. 

17. 

10  R. 

=  ?qr. 

8.    GO  units 

=  ?  doz. 

18. 

72  gr. 

=  ?  pwt. 

9.    3  qr. 

=  ?  sheets 

19. 

32  oz. 

(Av 

.)  =  ?ib. 

10.    54  cu. 

ft. 

=  ?  cu.  yd. 

20. 

4  score 

=  ?  units. 

214  ELEMENTS  OF   ARITHMETIC. 


REDUCTION  OP  DENOMINATE  NUMBERS, 

257.  The  process  of  changing  a  denominate  number  from 
one  denomination  to  another  without  altering  its  value  is 
called  Reduction. 

258.  The  process  of  changing  a  denominate  number  to 
an  equivalent  number  of  a  lower  denomination  is  called 
Reduction  to  Lower  Denominations,  or  Reduction  Descending, 

259.  The  process  of  changing  a  denominate  number  to 
an  equivalent  number  of  a  higher  denomination  is  called 
Reduction  to  Higher  Denominations,  or  Reduction  Ascending. 

REDUCTION  TO  LOWER  DENOMINATIONS. 

260.  1.    How  many  inches  are  there  in  7  yd.  1  ft.  4  in.  ? 

7 

3  Explanation.  —  Since  there   are  3  ft.  in  1  yd.,  in  7  yd. 

21     there  are  7  times  3  ft.,  or  21  ft.,  and  21  ft.  +  1  f t.  =  22  ft. 

1     Since  there  are  12  in.  in  1  ft.,  in  22  ft.  there  are  22  times  12  in., 

or  264  in.,  and  264  in.  +  4  in.  =  268  in.     Therefore  7  yd.  1  ft.  4 

in.  =  268  in. 

The  business  method  of  reducing  denominate  numbers  to 
264     lower  denominations  is  more  convenient  than  that  given  in  the 

4     explanation.      It  is  illustrated  by  the  process  in  the  margin. 

268 

261.  Hence  the  business  method  is  as  follows : 

To  reduce  denominate  numbers  to  loiver  denominations, 
multiply  the  number  of  the  highest  denomination  given  by  the 
7iumber  indicating  hoio  many  units  of  the  7iext  lower  denomi- 
nation are  equal  to  one  of  the  higher,  and  to  the  product  add 
the  number  given  of  this  lower  denomination. 

Proceed  in  like  manner  with  this  cmd  each  successive  result 
thus  obtained,  until  the  number  is  reduced  to  the  required 
denomination. 


22 
12 


DENOMINATE  NUiMBERS,  215 

Reduce  to  lower  denominations  : 

2.  4  yd.  2  ft.  8  in.  23.  4  rd.  3  yd.  2  ft.  6  in. 

3.  12  yd.  1  ft.  7  in.  24.  5  rd.  2  yd.  1  ft.  8  in. 

4.  15  yd.  2  ft.  10  in.  25.  8  rd.  1  yd.  2  ft.  10  in. 

5.  27  yd.  2  ft.  11  in.  26.  3  sq. yd. 4 sq.ft.  18 sq.  in. 

6.  6  gal.  3  qt.  1  pt.  27.  5 sq. yd. 8 sq.ft.  100 sq. in. 

7.  13  gal.  2  qt.  1  pt.  28.  2  A.  110  sq.  rd.  5  sq.  yd 

8.  15  gal.  1  qt.  1  pt.  2  gi.  29.  8  A.  25  sq.  rd.  4  sq.  yd. 

9.  18  gal.  3  qt.  1  pt.  3  gi.  30.  2cu.yd.l0cu.ft.85cu.in. 

10.  4  hr.  15  min.  15  sec.  31.  4cu.yd.6cu.  ft.  714cu.  in. 

11.  10  lir.  12  min.  18  sec.  32.  3  lb.  2  oz.  7  dr.  1  sc. 

12.  18  hr.  20  min.  35  sec.  33.  4  lb.  8  oz.  5  dr.  2  sc. 

13.  4  cwt.  20  lb.  8  oz.  34.  3  R.  10  qr.  18  sheets. 

14.  8  cwt.  75  lb.  12  oz.  35.  4  R.  8  qr.  10  sheets. 

15.  2  T.  5  cwt.  80  lb.  12  oz.  36.  3  lb.  6  oz.  10  pwt. 

16.  4  T.  7  cwt.  35  lb.  9  oz.  37.  5  lb.  10  oz.  3  sc. 

17.  4  lb.  6  oz.  18  pwt.  10  gr.  38.  6  da.  10  hr.  25  min. 

18.  8  lb.  4  oz.  10  pwt.  8  gr.  39.  8  bu.  3  pk.  7  qt.  1  pt. 

19.  5  lb.  5  oz.  15  pwt.  9  gr.  40.  8  rd.  4  yd.  2  ft.  6  in. 

20.  3  bu.  3  pk.  5  qt.  1  pt.  41.  30  bu.  3  pk.  6  qt. 

21.  5  bu.  2  pk.  7  qt.  1  pt.  42.  5  A.  140  sq.  rd.  3  sq.  yd. 

22.  6  bu.  1  pk.  4  qt.  1  pt.  43.  7  mi.  100  rd.  13  ft.  6  in. 

44.  Reduce  .75  yd.  to  feet  and  inches. 

„75 
3  Explanation. — Since  in  1  yd.   there  are 

2^25  3  ft.,  in  .75  yd.  there  are  .75  of  3  ft.  or  2.25  ft. 

^2  Since  in  1  ft.  there  are  12  in.,  in  .25  ft.  there 

are  .25  of  12  in.,  or  3  in. 

Therefore,  .75  yd.  =  2  ft.  3  in. 


3.00 

J5  yd.  =  2  ft.  3  in 


45.  I  cwt.       48.    f  yd.         51,    .4  cu.  yd.     54.    .73  wk. 

46.  I  rd.  49.    .6  bu.         52.    f  mi.  55.    .3125  gSi 

47.  -I  wk.        50.    .45  T.        53.   f  A.  56.    .475  yd. 


216  ELEMENTS   OF   ARITHMETIC. 


REDUCTION  TO  HIGHER  DENOMINATIONS. 

262.    1.   Keduce  235  pints  to  bushels,  pecks,  etc. 


235  pt. 


Explanation.  —  Since  there 
-\-\r7  f  -j  ^f  are  2  pt.  in  1  qt.,  in  235  pt.  there 
Ll        -t    *  are  as  many  quarts  as  2  pt.  are 


^^  P^-   ^  ^t-  contained  times  in  235  qt.,  or  117 


3  bu.  2  pk.  qt.  1  pt. 


235  pt.  =  3  bu.  2  pk.  5  qt.  1  pt.    ^,  ^^^^^^  ^^^'^  ^^^  ^  f  ^^  ,\^\^ 
^  jr  X        X        there  are  as  many  pecks  m  1 1  /  qt, 

as  8  qt.  are  contained  times  in  117  qt.,  or  14  pk.  5  qt. 

Since  there  are  4  pk.  in  1  bu.,  in  14  pk.  there  are  as  many  bushels 
as  4  pk.  are  contained  times  in  14  pk.,  or  3  bu.  2  pk. 

Therefore,  235  pt.  are  equal  to  3  bu.  2  pk.  5  qt.  1  pt. 

263.   Hence  it  is  evident  that : 

To  reduce  a  deyiominate  number  to  units  of  higher  denomina- 
tions, the  given  number  is  divided  by  the  number  {ndicating 
how  many  units  of  the  given  denomination  malie  one  of  the 
next  higher  denomination. 

We  proceed  in  like  manner  imth  this,  and  each  successive 
quotient,  till  the  whole  is  reduced  to  the  required  denomination. 

The  last  quotient,  with  the  remainders  {when  there  are  any) 
annexed,  will  be  the  required  answer. 

2.    Eeduce  3265  inches  to  rods,  yards,  etc. 

Suggestion.  —  In  dividing  by  5^  it  is 
usually  more  convenient  to  reduce  the 
divisor  to  halves  and  the  dividend  also 
to  halves  before  dividing. 

It  should  be  remembered  then,  that 
the  remainder,  if  there  is  any,  is  half- 
yards,  which  must  be  changed  to  yards. 
„. ^  —.J 

Reduce : 

3.  4213  gi.  to  gal.,  etc.  •  6.  3539  in.  to  rd.,  etc. 

4.  3185  pt.  to  gal.,  etc.  7.  2456  in.  to  rd.,  etc. 

5.  5724  gi.  to  gal.,  etc.  8.  1605  in.  to  rd.,  etc 


12 

3265  in. 

3 

272  ft.  1  in. 
90  yd.  2  ft. 
2 

11 

180    [or  2  yd. 
16  rd.  4  half-yd.. 

DENOMINATE   NUMBERS.  217 

9.  3564  pt.  to  bu.,  etc.  18.  9894  min.  to  da.,  etc. 

10.  4683  pt.  to  bu.,  etc.  19.  3943  sq.  in.  to  sq.  yd.,  etc. 

11.  3947  pt.  to  bu.,  etc.  20.  5937  sq.  in.  to  sq.  yd.,  etc. 

12.  5164  oz.  to  cwt.,  etc.  21.  1894  sheets  to  K.,  etc. 

13.  8127  oz.  to  cwt.,  etc.  22.  2315  sheets  to  K.,  etc. 

14.  5983  lb.  to  tons,  etc.  23.  7864  cu.  in.  to  cu.  ft.,  etc. 

15.  3795  lb.  to  tons,  etc.  24.  6754  gr.  to  lb.  Troy,  etc. 

16.  4285  hr.  to  wk.,  etc.  25.  8386  gr.  to  lb.  Troy,  etc. 

17.  8168  sec.  to  hr.,  etc.  26.  9138  gr.  to  lb.  Apoth.,  etc. 

27.  Keduce  2  qt.  1  pt.  to  the  fraction  of  a  gallon. 

Explanation.  —2  qt.  1  pt.  =  5  pt.  ;  1  gal.  =  8  pt.     Therefore  2  qt. 
1  pt.,  or  5  pt.,  are  f  of  1  gal.,  or  8  pt. 

28.  Keduce  1  ft.  3  in.  to  the  fraction  of  a  yard. 
29-    Reduce  3  qt.  1  pt.  to  the  fraction  of  a  peck. 

30.  Reduce  4  pk.  3  qt.  to  the  fraction  of  a  bushel. 

31.  Reduce  2  qt.  1  pt.  3  gi.  to  the  fraction  of  a  gallon. 

32.  Reduce  2  pk.  4  qt.  1  pt.  to  the  fraction  of  a  bushel. 

33.  Reduce  4  yd.  2  ft.  6  in.  to  the  fraction  of  a  rod. 


ADDITION  OP  DENOMINATE  NUMBERS. 

264.    1.    Find  the  sum  of  3  gal.  3  qt.  1  pt. ;  5  gal.  2  qt. 

1  pt.  ;  6  gal.  2  qt.  1  pt. ;  8  gal.  3  qt. 

Explanation.  —  The  numbers  are  written  so  that 
units  of  the  same  denomination  stand  in  the  same  col- 
umn, and  we  begin  at  the  lowest  denomination  to  add. 

The  sum  of  the  pints  is  3  pt. ,  which  are  equal  to  1  qt. 
1  pt.  ;  the  1  pt.  is  written  under  pints  and  the  1  qt.  re- 
served to  add  to  the  quarts. 
nA      o      ^  The  sum  of  the  quarts  is  11  qt.,  which  are  equal  to  2 

gal.  3  qt.  ;  the  3  qt.  are  written  under  the  quarts  and  the 

2  gal.  reserved  to  add  to  the  gallons. 

The  sum  of  the  gallons  is  2-1  gal.,  which  are  written  under  the 
gallons. 


gal. 

qt. 

pt. 

3 

3 

1 

5 

2 

1 

6 

2 

1 

8 

3 

218  ELEMENTS  OF  ARITHMETIC. 

Find  the  sum  of : 

2.  3  gal.  3  qt.  1  pt. ;  18  gal.  2  qt.  1  pt. ;  12  gal.  3  qt. 

3.  8  bu.  3  pk.  5  qt. ;  16  bu.  2  pk.  7  qt. ;  12  bu.  3  pk.  4  qt. 

4.  7  da.  5  hr.  20  miii. ;  8  da.  14  hr.  10  min.  ;  6  da.  18  hr. 

5.  24  lb.  6  oz.  (Av.)  ;  35  lb.  14  oz. ;  26  lb.  10  oz. ;  83  lb. 

6.  13  lb.  4  oz.  10  pwt.  8  gr. ;  11  lb.  5  oz.  11  pwt.  16  gr. ; 

10  lb.  10  oz.  15  pwt.  21  gr. 

7.  10  yd.  2  ft.  10  in. ;  15  yd.  1  ft.  9  in. ;  8  yd.  2  ft.  7  in. ; 
18  yd.  1  ft.  11  in. ;  16  yd.  2  ft.  8  in. 

8.  15  gal.  3  qt.  1  pt.  3  gi. ;  18  gal.  2  qt.  1  pt.  2  gi. ;  28 
gal.  2  qt. ;  21  gal.  1  pt.  3  gi. 

9.  12  A.  35  sq.  rd. ;  14  A.  110  sq.  rd. ;  15  A.  132  sq.  rd. 

11  A.  96  sq.  rd. ;  25  A.  100  sq.  rd. 

10.  5  T.  6  cwt.  14  lb.  10  oz.;   7  T.  15  cwt.  36  lb.  15  oz. 
17  T.  8  cwt.  84  lb.  12  oz. ;  10  T.  9  cwt.  94  lb.  11  oz. 

11.  5  yd.  2  ft.  8  in. ;   4  yd.  1  ft.  11  in. ;   4  yd.  2  ft.  9  in. 
3  yd.  1  f t. ;  4  yd,  8  in. ;  |  of  a  yard  and  |  of  a  foot. 

SUBTRACTION    OP    DENOMINATE    NUMBERS. 

265.   1.    From  12  lb.  6  oz.  7  pwt.  take  3  lb.  8  oz.  10  pwt. 

Explanation.  —  The  numbers  are  written  so  that 

units  of  the  same  denomination  stand  in  the  same 

cokmm,  and  we  begin  at  the  lowest  denomination  to 

8        9        17      subtract.     Since  10  pwt.  cannot  be  subtracted  from 

7  pwt.,  a  unit  of  the  next  higher  order  is  united  with 

the  7  pwt.,  making  1  oz.  7  pwt.,  or  27  pwt.    Then  27  pwt.  —  10  pwt.  = 

17  pwt.,  which  we  write  under  pwt.  in  the  remainder. 

Since  1  oz.  was  united  with  the  7  pwt.,  only  5  oz.  are  left  in  the 
minuend.  Inasmuch  as  8  oz.  cannot  be  subtracted  from  5  oz.,  a  unit 
of  the  next  higher  order  is  united  with  it,  making  1  lb.  5  oz.,  or  17  oz. 
Then  17  oz.  —  8  oz.  ==  9oz.,  which  we  write  under  oz,  in  the  remainder. 

Since  1  lb,  was  changed  to  ounces,  there  are  but  11  lb.  left  in  the 
minuend.     Then  11  lb.  -  3  lb.  =  8  lb. 

Therefore,  the  remainder  is  8  lb.  9  oz.  17  pwl. 

2.    From  8  bu.  4  pk.  3  qt.  take  3  bu.  2  pk.  7  qt. 


lb. 

oz. 

pwt. 

12 

6 

7 

3 

8 

10 

DENOMINATE   NUMBERS.  219 

3.  From  12  gal.  2  qt.  1  pt.  2  gi.  take  5  gal.  3  qt.  1  pt.  3  gi. 

4.  From  10  da.  5  hr.  20  min.  take  7  cla.  7  hr.  10  min. 

5.  From  31  lb.  4  oz.  15  pwt.  take  8  lb.  6  oz.  12  pwt. 

6.  From  15  yd.  2  ft.  7  in.  take  4  yd.  2  ft.  10  in. 

7.  From  12  rd.  4  yd.  1  ft.  take  8  rd.  4  yd.  2  ft. 

8.  From  13  lb.  8  oz.  5  dr.  take  7  lb.  10  oz.  7  dr. 

9.  From  18  A.  18  sq.  rd.  take  7  A.  110  sq.  rd. 

10.  From  40  sq.  yd.  8  sq.  ft.  51  sq.  in.  take  15  sq.  rd. 
6  sq.  ft.  100  sq.  in. 

11.  From  25  T.  8  cwt.   75  lb.  10  oz.  take  10  T.  11  cwt. 
35  lb.  15  oz. 

12.  How  many  years,  months,  and  days  intervened  be- 
tween Dec.  24,  1892,  and  May  26,  1842  ? 

Explanation.  —  Since  the  later  date  expresses 
the  greater  period  of  time,  it  is  written  as  the 
minuend,  and  the  earlier  date  is,  therefore,  the 
„Q     subtrahend.     In  writing  the  dates  the  number  of 
the  month  is  given  instead  of  its  name.     The  sub- 
traction is  then  performed  as  in  denominate  numbers,  considering  a 
month  as  30  days,  and  a  year  as  12  months. 


yr. 

mo. 

da. 

1892 

12 

24 

1842 

5 

26 

? 


13.  How  long  was  it  from  Dec.  10, 1835,  to  May  15, 188 

14.  How  long  was  it  from  June  13, 1864,  to  Jan.  1,  1890  ? 

15.  How  long  was  it  from  Apr.  8, 1839,  to  July  7,  1891? 

16.  How  long  was  it  from  Sept.  23, 1881,  to  Aug.  10, 1892  ? 


MULTIPLICATION  OP  DENOMINATE  NUMBERS. 

266.   1.  Multiply  3  bu.  3  pk.  7  qt.  by  7. 

Explanation,  —  7  times  7  qt,  are  40  qt.,  or  6  pk. 
1  qt.     The  1  qt.  is  written  under  qt.  in  the  product, 
and  the  0  pk.  reserved  to  add  to  the  pecks,     7  times 
~[       8  pk.  are  21  pk. ;  21  pk.  +  6  pk.  reserved  are  27  pk,, 
or  6  bu,  3  pk.     The  3  pk,  are  written  under  pk.  in 
the  product,  and  the  6  bu.  reserved  to  add  to  tlie  bushels. 

Seven  times  3  bu,  are  21  bu, ;  21  bu.  +  6  bu,  reserved  are  27  bu. 
Therefore,  the  product  is  27  bu.  3  pk.  1  qt. 


bu. 

pk. 

qt. 

3 

3 

7 
7 

220  ELEMENTS   OF  ARITHMETICo 

2.  Multiply  4  bu.  3  pk.  5  qt.  by  7 

3.  Multiply  5  gal.  1  qt.  3  pt.  2  gi.  by  6, 

4.  Multiply  5  lb.  6  oz.  10  pwt.  8  gr.  by  8. 

6.  Multiply  7  lb.  8  oz.  5  dr.  2  sc.  10  gr.  by  7, 

6.  Multiply  3  hr.  20  niiu.  35  sec,  by  5, 

7.  Multiply  2  T.  5  cwt.  48  lb.  15  oz.  by  8. 

8.  Multiply  2  rd.  3  yd.  2  ft.  10  in.  by  9. 

9.  Multiply  12  cu.  yd.  15  cu.  ft.  1115  cu.  in.  by 

10.  Multiply  8  sq.  yd.  2  sq.  ft.  45  sq.  in.  by  5. 

11.  Multiply  5  da.  8  hr.  15  min.  25  sec.  by  4. 

12.  Multiply  12  rd.  4  yd.  2  ft.  8  in,  by  8. 

13.  Multiply  5  R.  8  qr.  16  sheets  by  9. 

14.  Multiply  6  bar.  10  gal.  3  qt.  1  pt.  by  7, 


DIVISION  OP  DENOMINATE  NUMBERS. 

267.    1.  Divide  14  gal.  3  qt.  1  pt.  by  4. 

gal.      qt.      pt.  Explanation. —  One  fourth  of  14  gal.  is  3  gal. 

4)14  3  1  and  2  gal.  remainder.  The  3  gal.  are  written  ia 
3  2  1^  the  quotient,  and  the  2  gal.  remainder  are  united 
^      with  the  3  qt.,  making  11  qt. 

One  fourth  of  11  qt.  is  2  qt.  and  3  qt.  remainder.  The  2  qt.  are 
written  in  the  quotient,  and  the  3  qt.  remainder  united  with  the  1  pt. , 
making  7  pt. 

One  fourth  of  7  pt.  is  1|  pt. 

Therefore,  the  quotient  is  3  gal.  2  qt.  If  pt. 

2.  Divide  21  gal.  3  qt.  1  pt.  3  gi.  hy  6. 

3.  Divide  15  bu.  3  pk.  5  qt.  1  pt.  by  4. 

4.  Divide  13  yd.  2  ft.  6  in.  by  7. 

5.  Divide  23  cwt.  68  lb.  10  oz.  by  5. 

6.  Divide  42  lb.  8  oz.  15  pwt.  by  9. 

7.  Divide  22  lb.  7  oz.  5  dr.  2  sc.  by  8. 

8.  Divide  19  hr.  10  min.  36  sec.  by  6. 

9.  Divide  21  sq.^  yd.  7  sq.  ft.  45  sq.  in.  by  4, 
10.  Divide  15  rd.'4  yd.  2  ft.  8  in.  by  5. 


DENOMINATE  NUMBERS. 


221 


PRACTICAL  MEASUREMENTS. 

268.  To  compute  the  area  of  rectangular  surfaces. 

1.  What  is  a  square  inch?  Draw  a  figure  6  in.  long  and 
1  in.  wide  with  all  its  angles  equal.  How  many  sq.  in. 
does  it  contain  ? 

2.  If  the  figure  were  2  in.  wide,  how  many  sq.  in.  would 
it  contain?  How  many  if  it  were  3  in.  wide  ?  4  in.  wide  ? 
5  in.  wide  ?     6  in.  wide  ?  ^ 

3.  What  is  a  square  foot?  How  many  sq.  ft.  are  there 
in  a  strip  of  ground  12  ft.  long  and  1  ft.  wide,  with  all  its 
angles  equal  ?  How  many,  if  it  is  2  ft.  wide  ?  3  ft.  wide  ? 
4  ft.  wide  ?     5  ft.  wide  ?     6  ft.  wide  ? 

4.  How  many  sq.  ydo  are  there  in  a  walk  15  yd.  long 
and  5  yd.  wide,  having  its  angles  all  equal  ? 

269.  A  figure  that  has  four  straight  sides 
and  four  equal  angles  is  called  a  Rectangle. 

The  angles  of  a  rectangle  are  all  right  angles.  Uectaugie. 

270.  The  number  of  square  units  that  a  surface  contains 
is  called  its  Area. 

Thus  the  area  of  a  rectangle  4  in.  long 
and  2  in.  wide  is  8  sq.  in.,  for  it  may  be 
divided  into  2  rows,  each  containing  4  sq.  in. 

For  additional  instruction  in  measure- 
ments, consult  the  author's  Standard  Arithmetic. 


WRITTEN   EXERCISES. 
271.    1.    What  is  the  area  of  a  rectangular  walk  22  ft. 
long  and  5  ft.  6  in.  broad  ? 

Explanation.  —  Since  the  walk 
is  22  ft.  long  and  5i  ft.  wide,  its 
area  in  feet  is  121  sq.  ft. 

The  same  result  can  be  obtained 
by  expressing-the  length  and  breadth 
in  inches,  and  reducing  the  result 
obtained  to  square  feet. 


22x51  =  121 
or, 
264  x(j6  =  17,424 
17,424  sq.  in.  =  121  sq.  ft. 


222  ELEMENTS  OF  ARITHMETIC. 

272.   Hence  it  is  evident  that : 

The  area  of  any  rectangular  surface  is  obtained  by  finding 

the  product  of  the  numbers  expressing  its  length  a^id  breadth. 

The  length  and  breadth  must  be  expressed  in  units  of  the  same 
denomination. 

2.  What  is  the  area  of  a  rectangular  room  17  ft.  long 
and  23  ft.  wide  ? 

3.  How  many  sq.  ft.  of  surface  are  there  in  the  side  wall 
of  a  rof)m  16  ft.  long  and  12  ft.  high  ?     How  many  sq.  yd.  ? 

4.  The  floor  of  a  rectangular  room  18  ft.  wide  and  24 
ft.  long  contains  how  many  sq.  yd.  ? 

5.  A  farm  in  the  form  of  a  rectangle  has  a  length  of 
160  rd.  and  a  breadth  of  120  rd.  How  many  sq.  rd.  of  land 
does  it  contain  ?     How  many  acres  ? 

6.  How  many  acres  are  there  in  a  rectangular  field  60 
rd.  wide  and  80  rd.  long  ? 

7.  A  farmer  paid  $2.50  per  acre  for  plowing  a  rec- 
tangular field  that  was  2^  rd.  wide  and  45  rd.  long.  What 
was  the  entire  expense  of  plowing  the  field  ? 

8.  A  plasterer  received  23<^  per  sq.  yd.  for  plastering 
the  walls  and  ceiling  of  a  room  18  ft.  long,  15  ft.  wide,  and 
12  ft.  high,  no  deduction  being  made  for  the  openings. 
How  much  did  lie  receive  ? 

9.  How  much  wall  it  cost  to  carpet  the  floor  of  the 
room  with  carpeting  1  yd.  wide  at  85^  per  yd.  ? 

10.  The  farm  of  Mr.  Hudson  is  160  rd.  long  and  150  rd. 
wide,  and  he  purchases  an  adjoining  farm  160  rd.  long  and 
90  rd.  wide.     How  many  acres  has  he  after  the  purchase  ? 

11.  A  room  18  ft.  wide  and  24  ft.  long  was  carpeted  with 
carpeting  1  yd.  wide  at  $  1.25  per  yd.    How  much  did  it  cost  ? 

12.  The  length  of  a  field  was  80  rd.  and  its  breadth  was 
60  rd.  If  half  of  it  was  planted  with  corn  and  half  the 
remainder  with  potatoes,  how  many  acres  were  planted  to 
each  ? 


DENOMINATE   NUMBERS.   /  '  223 

273.  To  compute  the  volume  of  rectangular  sc 

1.  Wliat  is  a  cubic  inch  ?    A  cubic  foot  ?    A  cubic  yard  ? 

2.  How  many  cubic  feet  of  stone  are  there  in  a  rec- 
tangular block  of  stone  which  is  6  ft.  long,  1  ft.  wide,  and  1 
ft  thick? 

3.  How  many  cubic  feet  would  there  be  if  the  block 
were  2  ft.  Avide  ?  2  ft.  thick  ?  2  ft.  wide  and  2  ft.  thick  ? 
How  many  if  it  were  7  ft.  long,  2  ft.  wide,  and  2  ft.  thick  ? 

4.  How  many  cubic  feet  are  there  in  a  similar  block 
which  is  6  ft.  long,  3  ft.  wide,  and  1  ft.  thick  ?  In  a  block 
6  ft.  long,  3  ft.  wide,  and  2  ft.  thick  ?  In  a  block  6  ft. 
long,  3  ft.  wide,  and  3  f t„  thick  ?  In  a  block  6  ft.  long,  4  ft. 
wide,  and  3  ft.  thick  ?  In  a  block  6  ft.  long,  3  ft.  wide,  and 
4  ft.  thick?  In  a  block  3  ft.  Jong,  4  ft.  wide,  and  6  ft. 
thick  ? 

274.  The  number  of  solid  units 
that  a  body  contains  is  its  Volume 
or  Solid  contents 0 

The  solid  units  are  cubic  inches, 
cubic  feet,  cubic  yards,  etc. 

If  a  sohd  is  4  ft.  long,  3  ft.  wide,  and 
3  ft.  thick,  its  volume  is  36  cu.  ft. ,  for 
it  may  be  divided  into  3  blocks,  each  containing  12  cu.  ft.,  as  shown 
by  the  illustration. 


^^e 


^fP^^ 


1  ^ 

Cubic 

ft. 

> 

I  1^ 

Cubic 

ft. 

ilill^ 

Cubic 

ft. 

^ 

(1 


^. 


275.  A  pile  of  wood 
or  stone  8  ft.  long,  4  ft. 
wide,  and  4  ft.  high  is 
called  a  Cord. 

A  pile  that  is  1  ft. 
long,  4  ft.  wide,  and  4 
feet  high  is  called  a 
Cord  Foot. 

Stone  walls  are  sometimes  measured  by  the  Perch,  which 
contains  24.75  cu.  ft. 


224  ELEMENTS   OF   ARITHMETIC, 

WRITTEN    EXERCISES. 

276.  1.  How  many  cubic  feet  of  stone  are  there  in  a 
rectangular  block  8  ft.  long,  5  ft.  3  in.  wide,  and  3  ft.  6  in. 
thick? 

o       p-i        oi  -I  A  J  Explanation. — ^  Since   a  block  8  ft.  long, 

'f         2  I  ft.  wide,  and  1  ft.  thick  contains  8  cu.  ft.,  a 

block  5}  ft.  wide  and  1  ft.  thick  contains  5^  times  8  cu.  ft.,  or  42 
cu.  ft. 

Since  a  block  8  ft.  long,  5}  ft.  wide,  and  1  ft.  thick,  contains  42  cu. 
ft.,  a  similar  block,  that  is  3^  ft.  thick,  contains  3 J  times  42  cu.  ft.,  or 
147  cu.  ft. 

277.  Hence  it  is  evident  that : 

The  volume  or  solid  contents  of  a  rectangular  solid  are 
obtained  hy  finding  the  product  of  the  numbers  expressing  its 
length,  breadth,  and  thickness. 

The  length,  breadth,  and  thickness  must  be  expressed  in  units  of 
the  same  denomination. 

Find  the  volume  of  solids  with  dimensions  as  follows  i 

2.  Length  10  ft.,  breadth  7  ft.,  thickness  6  ft. 

3.  Length  9  in.,  breadth  11  in.,  thickness  8  in. 

4.  Length  5  yd.,  breadth  8  yd.,  thickness  10  yd. 

5.  Length  14  ft.,  breadth  10  ft.,  thickness  15  ft. 

6.  Length  8  ft.  6  in.,  breadth  8  ft.,  thickness  10  ft. 

7.  Length  12  ft.  4  in.,  breadth  9  ft.,  thickness  8  ft. 

8.  Length  10  ft.  6  in.,  breadth  10  ft.,  thickness,  12  ft 

9.  Length  12  ft.  4  in.,  breadth  10  ft.,  thickness  8  ft. 

10.  Length  8  ft.  4  in.,  breadth  10  ft  6  in.,  thickness  8  ft. 

How  many  cords  of  wood  are  there  in  the  following  piles  ? 

11.  Length  24  ft.,  breadth  4  ft.,  height  4  ft 

12.  Length  32  ft.,  breadth  4  ft.,  height  8  ft. 

13.  Length  48  ft.,  breadth  16  ft,  height  12  ft. 

14.  How  much  will  the  wood  in  the  last  pile  cost  at 
^  5.50  per  cord  ? 


DENOMINATE   NUMBERS.  225 

15.  How  much  will  it  cost  to  excavate  a  cellar  which  is 
40  ft.  long,  22  ft.  wide,  and  8  ft  deep,  at  30j^  per  cubic  yard  ? 

16.  How  much  will  be  the  expense  of  laying  a  stone  wall 
40  ft.  long,  1  ft.  6  in.  thick,  and  6  ft.  high,  at  $  2.50  per 
perch  ? 

278.  To  measure  lumber. 

In  measuring  lumber,  boards  1  inch  thick  or  less  are 

estimated  by  the  number  of  square  feet  of  surface. 

Thus  a  board  1  foot  wide  and  10  feet  long  contains  10  feet  board 
measure,  if  it  is  1  inch  or  less  in  thickness. 

Tlie  length  and  width  must  both  be  expressed  in  feet. 

279.  When  lumber  is  more  than  1  inch  in  thickness,  the 
number  of  feet  board  measure  is  obtained  as  follows : 

The  length  in  feet  is  multiplied  by  the  breadth  in  feet,  and 

this  X)roduct  by  the  number  of  inches  in  thickness. 

Thus,  the  number  of  feet  board  measure  in  a  timber  12  ft.  long,  15 
in.  wide,  and  2^  in.  thick  is  obtained  as  follows: 

12  X  1^  X  2^  =:  37],  tlie  number  of  feet  board  measure. 

How  many  feet  are  there  in  the  following  boards : 

1.  16  ft.  long,  18  im  wide.      4.    20  ft.  long,  14  in.  wide. 

2.  12  ft.  long,  15  ill.  wide.      5.    15  ft.  long,  10  in.  wide. 

3.  18  ft.  long,    9  in.  wide.      6.    20  ft.  long,  16  in.  wide. 

7.  What  will  be  the  cost  of  10  boards,  each  12  ft.  long 
and  15  in.  wide,  at  $  20  per  thousand  feet  ? 

8.  How  many  feet  board  measure   are  there  in  20  joists 
18  ft.  long,  16  in.  wide,  and  3  in.  thick  ? 

9.  What  will  those  joists  cost  at  $  18  per  thousand  feet  ? 

10.  What  will  be  the  cost  of  20  planks  16  ft.  long,  14  in, 
wide,  and  2  in.  thick,  at  f  22  per  thousand  feet  ? 

11.  What  will  be  the  cost  of  20  planks  12  ft.  long,  15  in. 
wide,  and  1^  in.  thick,  and  40  scantlings  6  in.  wide,  12  ft. 
long  and  4  in.  thick,  at  $  18  per  thousand  feet  ? 

EL.     DF   AR.  —  15 


INTEREST. 


280c  When  a  person  borrows  money,  he  usually  has  to 
pay  a  certain  number  of  hundredths,  or  a  certain  per  cent^  of 
the  sum  for  the  use  of  it  for  each  year. 

281.  The  character  %  is  often  used  instead  of  the  words 
hundredths  or  per  cent. 

Thus,  5  hundredths,  or  5  per  cent,  is  expressed  5%. 

ORAL   EXERCISES. 

282.  1.  When  a  sum  equal  to  6%  of  the  money  loaned 
is  paid  for  the  use  of  it  for  each  year,  how  much  is  paid  for 
the  use  of  f  100  for  1  yr.,  or  what  is  the  interest  for  1  yr.  ? 

2.  When  the  rate  of  interest  is  5%  per  year,  how  much 
wall  be  the  interest  of  $  100  for  1  yr.  ?  For  2  yr.  ?  For  3  yr.  ? 

3.  When  the  rate  of  interest  is  8%  per  year,  what  will 
be  the  interest  of  $  100  for  1  yr.  ?     ^  200  ?     $  500  ? 

4.  When  the  rate  of  interest  is  6%  per  year,  what  will 
be  the  interest  of  $  400  for  1  yr.  ?  For  2  yr.  ?  For  2^  yr.  ? 
For  i  yr.  ? 

6.  When  the  rate  of  interest  is  4%  per  year,  how  much 
will  $  200  amount  to  in  1  yr.  ?     In  2  yr.  ?     In  2-1-  yr.  ? 

6.  When  $400  is  loaned  for  3  yr.  at  5%  per  year,  how 
much  is  the  interest  ?     What  is  the  amount  due  ? 

What  is  the  interest  of : 

7.  $  600  for  1  yr.  at  5%  ?  d.    $  800  for  2  yr.  at  4%  ? 

8.  $500for  1  yr.  at  C%  ?  ]0.    $400  for  3  yr.  at  5%  ? 

226 


INTEREST.  227 

11.  $600forliyr.at3%?  15.  $800  for  11  mo.  at  6%  ? 

12.  $200for2iyr.  at5%?  16.  $600  for  1  mo.  10  da.  at  6% ? 

13.  $800foriyr.at6%?  17.  $600  for  2  mo.  10  cla.  at  6% ? 

14.  $  600  fori  mo.  at  6%?  18.  $800  for  Syr.  at  5%  ? 

283.  The  sum  paid  for  the  use  of  money  is  called  Interest, 

284.  The  sum  for  the  use  of  which  interest  is  paid  is 
called  the  Principal. 

285.  The  sum  of  the  principal  and  interest  is  called  the 
Amount. 

286.  In  computing  interest  it. is  usual  to  regard  a  year 
as  12  months,  and  a  month  as  30  days. 

WRITTEN    EXERCISES. 

287.  1.    What  is  the  interest  of  $  420.50  for  5  yr.  at  5%  ? 

Solution. 
$420.50  Principal. 
.05  Rate. 


$21.0250  Interest  for  1  yr. 
5 


$105.1250  Interest  for  5  yr. 
Find  the  interest  of : 

2.  $212.34  for  3  yr.  at  4%.     5.  $219.20  for  4  yr.  at  7%. 

3.  $426.12  for  4  yr.  at  5%.     6.  $324.16  for  5  yr.  at  4%. 

4.  $  324.25  for  5  yr.  at  6%.     7.  $  582.30  for  6  yr.  at  5%. 
8.    What  is  the  interest  of  $  366.20  for  3  yr.  8  mo.  at  6%? 

Solution. 

$366.20 
.06 


$21.9720,  int.  for  1  yr. 

3 

$65.9160,  int.  for  3  yr. 
I  of  the  int.  for'l  yr.    =    10.9860,  int.  for  6  mo. 

I  of  the  int.  for  6  mo.  =      3.6620,  int.  for  2  mo. 

$80.5640,  int.  for  3  yr.  8  mo. 


228  ELEMENTS  OF   ARITH^METIC. 

Find  the  interest  of : 

9.  $412.20  for  2  yr.  8  mo.  at  4%. 

10.  $  318.36  for  3  yr.  6  mo.  at  6%. 

11.  $425.24  for  4  yr.  5  mo.  at  5%. 

12.  $  685.60  for  3  yr.  7  mo.  at  7%. 

13.  $437.28  for  5  yr.  2  mo.  at  8%. 

14.  $  314.26  for  4  yr.  4  mo.  at  3%. 

15.  $  736.16  for  2  yr.  10  mo.  at  6%. 

16.   Find  the  amount  of  $  524.36  for  2  yr.  5  mo.  18  da. 

at  7%. 

Solution. 

$524.36 
.07 


$36.7052,  int.  for  1  yr. 
2 

$  73.4104,  int.  for  2  yr. 
1  of  the  int.  for  1  yr.  =     12.2350,  int.  for  4  mo. 

I  of  tlie  int.  for  4  mo.  =      3.0587,  int.  for  1  mo. 

I  of  the  int.  for  1  mo.  =      1.5293,  int.  for  15  da. 

A  of  the  int.  for  15  da.  =        .3058,  int.  for  3  da. 

$  90.5392,  int.  for  2  yr.  5  mo.  18  da. 
524.36     ,  principal. 

$614.8992,  amount. 
Find  the  interest  and  amount  of : 

17.  $360.24  for  2  yr.  7  mo.  15  da.  at  5%. 

18.  $415.48  for  3  yr.  5  mo.  10  da.  at  6%. 

19.  $217.36  for  2  yr.  6  mo.  15  da.  at  4%. 

20.  $423.35  for  3  yr.  8  mo.  15  da.  at  6%. 

21.  $720.16  for  3  yr.  7  mo.  17  da.  at  5%. 

22.  $468.36  for  3  yr.  9  mo.  10  da.  at  4%. 

23.  $219.47  for  2  yr.  5  mo.  12  da.  at  6%. 

24.  $325.34  for  1  yr.  1  mo.  13  da.  at  7%. 

25.  $436.33  for  4  yr.  2  mo.  10  da.  at  5%. 

26.  $297.45  for  5  yr.  3  mo.  15  da.  at  6%. 


GENERAL   REVIEW, 


288.   lo   What  will  50  sheep  cost  if  75  are  worth  |?375? 

2.  How  many  years  is  it  since  the  Declaration  of  Inde- 
pendence was  signed  ? 

3o  A  rope  13  yd.  2  ft.  6  in.  long  was  cut  into  3  equal 
parts.     How  long  were  they  ? 

4.  A  farmer  sold  at  one  time  3  bu.  3  pk.  7  qt.  of  clover 
seed,  and  at  another  5  bu.  2  pk.  6  qt.  How  much  did  he 
sell? 

5o  A  wagon  wheel  is  15  ft.  8  in.  in  circumference.  How 
many  revolutions  \vill  it  make  in  going  5  miles  ? 

6.  James  spent  57  cents,  which  was  -|  of  all  the  money 
he  had.     How  much  money  had  he  ? 

7.  A  father  gave  his  children  $8  apiece.  If  he  had 
given  them  $12  apiece  it  would  have  taken  $40  more. 
How  many  children  were  there  ? 

8.  The  larger  of  two  fractions  is  \^,  and  the  difference 
between  them  is  f .     What  is  the  smaller  fraction  ? 

9.  If  I  of  an  acre  of  land  produces  125  bushels  of 
potatoes,  how  many  bushels  will  3^  acres  produce  ? 

10.  A  can  do  a  piece  of  work  in  10  days,  and  B  can  do  it 
in  8  days.     In  what  time  can  both  do  it  ? 

11.  A  man  who  owned  -f  of  a  factory  sold  |  of  his  share 
for  $  3500.     What  w^as  the  factory  worth  at  that  rate  ? 

12.  Two  boys  did  some  work  for  65  cents.  If  one  of 
them  earned  15  cents  more  than  the  other,  how  much  did 
each  earn  ? 

220 


230  ELEMENTS   OF   ARITHMETIC. 

13.  A  man  who  spent  ^  of  his  money  and  ^12  more 
found  that  he  had  $  15  left.     How  much  had  he  at  first  ? 

14.  A  man  died  leaving  to  his  3  children  a  farm  of  317  A. 
115  sq.  rd.  to  be  divided  equally  among  them.  What  was 
each  one's  share  of  the  land  ? 

15.  How  many  acres  of  land,  at  $45  per  acre^  can  I  get 
^or  90  oxen  at  $37.75  per  head  ? 

16.  Mr.  David  Bounds  purchased  from  D.  Henderson  & 
Co.  13  yd.  muslin  @  8^,  8  yd.  sheeting  @  $  .111,  25  yd. 
ribbon  @  7^  per  yd.,  10  lb.  cotton  batting  at  8^^,  2  pr. 
gloves  @  $  1.25,  3  doz.  pearl  buttons  @  $  .27.  Make  out  a 
receipted  bill. 

17.  How  long  will  it  take  a  person  to  count  $1,000,000 
if  he  counts  $  5  per  second  for  10  hours  per  day  ? 

18.  If  a  railroad  train  runs  at  the  rate  of  521  nii.  per 
hour,  how  far  will  it  run  in  8  hr.  40  min.  ? 

19.  How  many  feet  board  measure  are  there  in  45  joists, 
each  18  ft.  long,  16  in.  wide,  and  2  in.  thick? 

20.  A  farmer  sold  3  T.  18  cwt.  85  lb.  of  hay  at  $  9.50  per 
ton.     Hew  much  did  he  get  for  it  ? 

21.  A  cask  of  wine  contained  5S  gab  3  qt.  1  pt.  How 
much  was  it  worth  at  $  2.75  per  gallon  ? 

22.  A  silversmith  sold  some  silver  plate  at  $  1.50  per 
ounce,  receiving  for  it  $  93.     How  much  did  he  sell  ? 

23.  A  merchant  had  320  yd.  of  cloth,  which  he  made 
into  suits  which  required  6  yd.  2  ft.  each.  How  many  suits 
did  he  make  ? 

24.  What  will  be  the  expense  of  painting  a  gable  roof 
48  ft.  long  and  25  ft.  wide  at  $  .30  per  sq.  yd.  ? 

25.  How  many  yards  of  matting,  1|-  yd.  wide,  will  be  re- 
quired to  cover  a  floor  7  yd.  2  ft.  7i  in.  wide,  and  18  ft.  long  ? 
What  will  be  the  cost  of  it  at  $  .70  per  lineal  yd.  ? 

26.  A  man  has  a  lot  containing  10  acres  that  is  25  rods 
wide.     How  hm":  is  it  ? 


GENERAL   REVIEWo  231 

27.  A  money-lender  loaned  $520  for  two  years  at  6%. 
How  much  money  should  he  receive  as  principal  and  interest 
at  the  end  of  the  time  ? 

28.  One  number  is  ^  of  another,  and  their  sum  is  490. 
What  are  the  numbers  ? 

29.  A  man  who  owned  f  of  a  vessel  sold  ^  of  his  share 
for  $  12,500.     At  that  rate,  what  was  the  vessel  worth  ? 

30.  Two  men  engaged  in  business,  one  furnishing  4  of 
their  capital,  and  the  other  the  rest.  If  they  gained  $  1477, 
what  was  each  one's  share  of  the  gain  ? 

31.  Mr.  Davis  loaned  his  nephew  $  .350  on  Jan.  1,  1890, 
at  6  %  interest.  If  the  debt  was  paid  Jan.  1,  1893,  how 
much  was  the  amount  ? 

32.  How  much  will  it  cost  to  carpet  a  room  18  ft,  wide 
and  24  ft.  long  at  $  1.15  per  sq.  yd.  ? 

33.  A  druggist  sold  1  lb.  of  quinine,  which  was  made  into 
two-grain  pills.     How  many  pills  were  there  ? 

34.  A  stationer  bought  10  reams  of  paper  at  $  1.90  per 
ream,  and  sold  it  at  1  cent  per  sheets  How  much  was  his 
gain  ? 

35.  Henry  bought  at  a  grocer's  15  lb.  12  oz.  of  butter  at 
32  cents  per  pound.     How  much  did  it  cost  him  ? 

36.  Two  brothers  earned  during  their  vacation  $  110,  but 
one  of  them  earned  only  |-  as  much  as  the  other.  How  much 
did  each  earn  ? 

37.  If  I  of  a  bbl.  of  flour  costs  $4.20,  what  will  7  bbl. 
cost  ? 

38=  Divide  I  of  I  of  I  of  II  of  3^  by  I  of  if  of  |  of  f 
of  9. 

39.  Sound  travels  about  1140  feet  per  second.  If  a  flash 
of  lightning  is  seen,  but  the  thunder  accompanying  it  is  not 
heard  until  35  seconds  later,  how  far  away  is  the  lightning  ? 

40.  The  product  of  two  numbers  is  3.35,  and  one  of  them 
is  .25.     What  is  the  other  number  ? 


ANSWERS, 


Page90.— 2.  283.  3.254.  4.231.  5.212.  6.136.  7.185c 
8.  300.  9.  198.  10.  198.  11.  280.  13.  315.  13.  274.  14.  213. 
15.  152.  16.  107.  17.  260.  18.  226.  19.  289.  20.  233.  21.  313. 
22.  293.     23.  197.     24.  179.     25.  271. 

Page  91.  — 26.  40  mi.  27.^57.  28.  88  yr.  29.  100  articles. 
30.  38  boys;  41  girls  ;  79  children.  31.  134  fishes.  32.  180  pounds. 
33.  262  pages.  34.  168  acres.  35.  31  mi.  36.  44  examples.  37.  79^. 
38.  87  bu. 

Page  92.  — 39.  50  chickens.  40.  $57.  41.  9Sf.  42.  $1.32„ 
43.    105  cows.  44.  $  1-.83.  45.   112  things.         46.  126  marbles. 

47.  97  leaves.     48.   121  loaves.    49.  262  qt.     50.   148  lb. 

Page  93.  — 51.  88  animals.  52.  $25.  53.  $40.  54.252.  55.266 
yd.  56.  .$88,  horse;  $176,  together.  57.  $  1.30.  58.  123  lines. 
59.  $  1821.     60.  1335  acres.     61.  3198  pupils  in  all  ;   1401,  4th  school. 

Page94.— 62.  593.  63.470.  64.541.  65.467.  66.461.  67.365. 
68.  542.  69.  5116.  70.  5048.  71.  3681.  72.  4129.  73.  5415. 
74.  5311.         75.  71,425.         76.  67,053.         77.  64,794.  78.  48,551. 

79.  23,437. 

Page  96.— 2.  532.  3.  723.  4.  351.  5.  413.  6.  214.  7.  441. 
8.  633.       9.  201.       10.  515.      11.  543.      12.  410.      13.  662.      14.  126. 

15.  456.      16.  634.      17.  613.      18.  245.       19.  724.      20.  654. 

Page  97.  — 24.  17.  25.  17.  26.  37.  27.  23.  28.  41.  29.  38. 
30.  16.  31.  23.  32.  54.  33.  24.  34.  34.  35.  48.  36.  3.  37.  68. 
38.  55.  39.  37.  40.  614.  41.  414.  42.  222.  43.  27.  44.  513. 
45.  223.  46.  110.  47.  208.  48.  129.  49.  126.  50.  317.  51.  428. 
52.  781.     53.  292.     54.  221. 

Page  98.  — 55.  166.  56.323.  57.468.  58.227.  59.414.  60.601. 
61.  209.  62.  153.  63.  625.  64.  434.  65.  657.  66.  609.  67.  641. 
68.  $49.  69.  $721.  70.  169  children.  71.  281  egg.s.  72.  758  birds. 
73.  79  pints.  74.  115  horses.  75.  265  cows.  76.  $^432.  77.  531  bu. 
78.  635  apples.     79.  365  da.     80.  56  yr.     81.  $  148.     82.  $  188. 

Page  99.  — 1.  534  gal.  2.  545  mi.  3.  3621  bricks.  4.  Lost  $22. 
5.  594.  6.  $  195.  7.  221  mi.  8.  Answers  differ.  9.  836  yd. 
10.  $  5569,  entire  cost ;  $2375  more.       11.  Answers  differ.     12.  112  ft. 

Page  100.  — 13.  $289.     14.  451  apple  trees ;  715  trees.     15.  34  bu„ 

16.  $97.49.     17.  $2.87.     18.  $89.     19.  $2058;  $431.50.     20.  $1262. 
21.  237  mi.     22.  613  mi.     23.  108  pupils.     24.  $556.     25.  580^. 

Page  101. —26.  375.  27.  225  pages.  28.  $  1.00.  29.  430  pupils. 
30  207  b;i.      31.  $3.25.      32.  $101.11.      33.   $351.(i0.      34.  $  104.64. 

232 


ANSWERS.  233 

35.  $90.64.  36.  .$11.65.  37.  $34.99.  38.  1 86.23.  39.  $44.73. 
40.  $109.95.     41.  $77.91. 

Page  107.  — 2.  230.  3.  222.  4.  196.  5.  558.  6.  602.  7.  392. 
8.  207.  S.  464.  10.  387.  11.  510.  12.  957.  13.  1104.  14.  2905. 
15.  2976.       16.  4230.       17.   1456.       18.  2754.       19.  1625.      20.  7776. 

21.  1300.     22.  4896.      23.  6524.      24.  5709.      25.   10,044.      26.  12,980. 

27.  14,305.        28.  11,826.        29.  34,122.        30.  30,672.         31.  21,116. 

32.  73,512.         33.  27,538.         34.  27,340.         35.  0,346.         36.  47,124. 

37.  23,844.  38.  26,400  ft.  39.  195  yd.  40.  $187.50.  41.  $66.00. 
42.  376  mi.     43.  $  426.25.     44.  $  190.80. 

Page  108.— 45.  $630.00.  46.  $91.75.  47.  $  58,600.  48.  $25,722.00. 
49.  $71,667.  50.  50,592  papers.  51.  $698.80.  52.  $832.95.  53.1000 
tons.  54.  230,370  passengers.  55.  $269,465.  56.  $1921.35.  57.83,208. 
58.  22,648  mi. 

Page  109.— 59.  68,355  bmiches.  60.  11,738;  17,607;  23,476; 
29,345;  35,214;  41,083;  46,952;  52,821.  61.  14,592;  21,888;  29,184; 
36,480;  43,776;  51,072;  58,368;  65,664.  62.  16,328;  24,492;  32,656; 
40,820  ;  48,984  ;  57,148  ;  65,312  ;  73,476.  63.  7908  ;  11,862  ;  15,816  ; 
19,770;  23,724;  27,678;  31,632;  35,586.  64.  7714;  11,571;  15,428; 
19,285;  23,142;  26,999;  30,856;  34,713.  65.  11,358;  17,037;  22,716; 
28,395;   34,074;    39,753;   45,432;    51,111.  66.   173,946;   260,919; 

347,892;  434,865;  521,838;  608,811;  695,784;  782,757.  67.  47,094; 
70,641;  94,188;  117,735;  141,282;  164,829;  188,376;  211,923. 
68.  130,264;  195,396;  260,528;  325,660;  390,792;  455,924;  521,056; 
586,188.  69.37,144;  55,716;  74,288;  92,860;  111,432;  130,004; 
148,576;  167,148.  70.  84,728;  127,092;  169,456;  211,820;  254,184; 
296,548;  338,912;  381,276.  71.  76,906;  115,359;  153,812;  192,265; 
230,718;  269,171;  307,624;  346,077.  72.  96,724;  145,086;  193,448; 
241,810;  290,172;  338,534;  386,896;  435,258.  73.  65,146;  97,719; 
130,292;  162,865;  195,438;  228,011;  260,584;  293,157.  74.  143,370; 
215,055;  286,740;  358,425;  430,110;  501,795;  573,480;  645,165. 

Page  110. —1.  450.  2.670.  3.380.  4.460.  5.760.  6.32,400. 
7.  41,600.  8.  72,300.  9.  34,200.  10.  51,800.  11.  5,463,000. 
12.  3,784,000.       13.  51,690,000.       14.  42,370,000.       15.  819,300,000. 

Pagelll.  — 17.  900.     18.  1110.      19.  2720.      20.  5840.     21.  3360. 

22.  3600.     23.  1500.      24.  2070.      25.  5880.      26.  3900.      27.  173,000. 

28.  83,700.      29.  302,600.      30.  237,600.       31.  154,000.       32.  298,800. 

33.  272,800.     34.  191,400.     35.  345,500.     36.  177,600.     37.  23.676,000. 

38.  21,116,000.  39.  58,555,000.  40.  65,034,000.  41.  43,096,000. 
42.  36,970,000.  43.  8,607,000.  44.  19,998,000.  45.  35,060,000. 
46.  55,092,000. 


Page  112.  - 

-51.  6408.        52. 

22,572. 

53.  55,998. 

54.  11,742. 

55.  12,555. 

56.  40,260.         57. 

51,471, 

58.  33,432. 

59.  12,354. 

60.  69,768. 

61.   70,308.        62. 

22,240. 

63.  59,616. 

64.  45,936. 

65.  16,016.     66 

.  907,420.     67.  2,256,404. 

68.  2,237,340.     69.  3,255,112. 

70.    1,640,536. 

71.    3,642,690. 

72. 

1,062,152.         73. 

1,330,329. 

74.   6,780,025. 

75.   6,938,840. 

76. 

2,883,686.         77. 

8,189,100. 

78.    5,836,005. 

79.   4,965,954. 

80 

3,468,255.         81. 

4,512,816. 

82.   2,749,835. 

83.   5,996,046. 

84. 

4,323,284.         85. 

2,673,054. 

86.   4,84-6,122. 

87.   2,353,176. 

88. 

27,381,900.       89. 

29,277,136. 

234  ELEMENTS   OF   ARITHMETIC. 

90.  60,355,476.  91.  302,422,875.  92.  468,083,374.  93.  186,173,400, 
94.  307,907,152.     95.  447,832,404. 

Page  113. —  96.  $28,189.68.  97.  #14,795.60.  98.  $52,071.30. 
99.  $71,125.30.  100.  $29,647.02.  101.  $2,164,284.20.  102.  $2,053,- 
983.00.  103.  $2,884,098.15.  104.  $  1,450,457.60.  105.  $2,774,732.64. 
106.  $2808.65,  107.195,360  ft.  108.  $92,022.  109.  441,066  bricks. 
110.  $16,686.42.  111.  $516.46.  112.  14,208  hides.  113.  $4847.04. 
114.  49,680  pencils.  115.  525,600  min.  116.  $  3538.75.  117.  207,360 
times.     118.   3,747,090. 

Page  114.  —  119.  11,341,784  gal.  120.  $ 6,356,220.  121.  $  444,475. 
122.  160,965  bbl.     123.  $75,209.40. 

Page  116.  — 1.  $220.  2.  $16,675.  3.  $950.  4.  $8024.  5.  $321.50. 
6.  $184.81.     7.  A,  $4918;  B,  $4295. 

Page  117.  — 8.  $243.25.  9.  $653.75.  10.  $3-32.75.  11.  $  10,136.50. 
12.  $1924.25.  13.  $11,340.  14.  $8280.  15.  $77  loss.  16.  $32,810. 
17.  $773.70.     18.  $6840. 

PagellS.  — 19.  $2526.75.  20.  $31,298.75.  21.  $12,799.60. 

22.  1740  people.  23.  $  13,833.  24.  10,720,000  seeds.  25.  252,018,000 
fishes.     26.  147  mi.    27.  123  mi. 

Page  123.  —6.  1218.  7.  1366.  8.  594.  9.  496.  10.  1545.  11.  984. 
12.   1078.     13.  876.     14.  454.     15.  166.     16.  125.     17.  211.     18.  554. 

19.  1353.  20.  518. 

Page  124.  — 21.  1645^.    22.  913 J.   23.  $14.88|.   24.  $9.08|. 

25.  $10,428.  26.  $14,653.  27.  $8374f.  28.  $22961-.  29.  $  12,052. 
30.  $4857|.  31.  $9479i.  32.  $10,149.  33.  $12,024.  34.  $2862/^ 
35.  454.  36.  237.  37.  391.  38.  459.  39.  13,170|.  40.  17,897. 
41.  4390j3...  42.  450  coats.  43.  211  weeks.  44.  $753.  45.  156  hr. 
46.  392  hr.  47.  689  people.  48.  3271  yd.  49.  6523  cloaks.  50.  586 
fathoms.  51.650;  487| ;  390;  325;  278f  ;  243 1 ;  216%.  52.492;  369; 
2951;  246;  210f  ;  184f  ;  164.  53.  1453i  ;  1090;  872;  726i ;  622f- ; 
545  ;  484|.  54.  981  ;  735| ;  588| ;  490| ;  420f  ;  367| ;  327.  55.  1872  ; 
1404  ;  1123J  ;  936  ;  8021  ;  702  ;  624.  56.  230.3  ;  1727^  ;  13814  ;  llolf  ; 
987;  863|;  767f.  57.  872;  654;  523 i ;  4-36;  373f ;  327;  290|. 
58.  628;  471;  376f  ;  314;  2691;  2.35^;  209|.  59.  5762;  4321 1  ; 
3457i  ;  2881  ;  24692  ;  2160| ;  1920f,.  60.  3871| ;  2903| ;  2323  ;  1935^  ; 
16592;  1451|;  1296|.   61.  14,951';  11,213|;  8970| ;  7475| ;  6407f ; 

If.  62.  6269f  ;  4702^  ;  37614  ;  3134| ;  2687  ;  2351 1 ;  2089|. 
55,444f;  46,203f  ;  39,603f;  34,652|  ;  30,802|. 

64.  182,779  ;  137,084i  ;  109,667f  ;  91,389| ;  78,333f ;  68,542,^  ;  60,926|. 

65.  129,882|;  97,412;  77,929| ;  64,941| ;  55,664;  48,706;  43,294|. 

66.  87,3162;  65,487f ;  52,.390 ;  43,658| ;  37,421f;  32,743| ;  29,105|. 
Pagel26.  —  4.  372.   5.486.   6.387^^.   7.  426i&^.   8.384^2^-. 

9.  573/^0-  10.  425Ji>,.  11.  298^^^',.  12.  342x§o.  13.  ^IS^^- 
14.  52,8^,.   15.  32/gVo-   16-  Q^^Wo-    17-  54j-V\fj.   18.  69rWo- 

20.  3411;].  21.  2433V  22.  81^^.  23.  49/^.  24.  329/^0.  25.  95|2i. 

26.  58^^.  27.68IH-  28- 1<>«M&-  29.39/^.  30.  96|ff .  31.135m. 
Page  128.-36.  382.  37.  213.  38.  m.     39,  48,  40.  55,  41.  73. 


A^^SWERS.  235 

42.  456.  43.  144.  44.  07.  45.  82.  46.  37.  47.  36.  48.  1829if ; 
1200P;  893||;  711^;  691i§.  49.  SllS^^;  2046i-|;  1522||  ;  1212f f ; 
1007||.  50.  4507/'r;  2957||;  2201^0;  1752i| ;  1456if.  51.  2175.y2j-; 
1427i| ;  1062^1 ;  8"454t ;  702i|.  52.  4034,^  ;  2647f  I ;  1970|| ;  15693''^  ; 
1303||.  53.  3054^2.;  2004/.,;  1491||;  li87||;  986ff.  54.  4755.^^^; 
312I322  ;  232211 ;  i849|| ;  1536|i.  55.  979^\  ;  642p  ;  478/3  ;  3801| ; 
316||.  56.  338611;  22221 1;  1653|| ;  1316f|;  lOO^V  57.  1928^; 
126511;  941i|;  750^^;  623ia,  58.  1704^f- ;  1118i?  ;  832i| ;  662f  | ; 
550|f.  59.  646^^;  424 32^  ;  315|| ;  251||  ;"'208|S.  60.  4237i|;  2781; 
206911;  1648;  1369^^3.  ^61.  2229.ii;  14633-^;  1088|| ;  867 i- ;  720|f. 
62.  4566^^  ;  2996|| ;  2230^^3  ;  1775i| ;  1475||."'  63.  33,515,*^  ;  21,994,i| ; 
16,367|f  ;  13,033|^|;  10,828^5-  64.  23,453 .^ f  ;  15,391|^;  ll,454|f  ; 
9120if;  7577||.  65.  31,859./r;  20,907f|;  15,559^^/;  12,.389|^ ; 
10,2936V  66.  35,697 Jf;  23,426/^;  17,433|-|  ;  13,882l|;  ll,532|f, 
67.  39,9202V;  26,19711;"  19,495i|;  i5,524fi-;  12,897|f.  68.  1024i-f; 
13532-i;  1995;  S2A^\;  1083„  69.  970fi.;  12821  f;  1890 ^V  ;  780|i; 
1026iv  70.  266511;  3522i|;  5191t\;  2144Jf;  28I83V  71.  1302|4  ; 
172111;  2537^5^;  1048;  1377if.  72.  8O63V;  1065.^38  ;  1569i| ;  648i| ; 
8523V   73.  1374/^;  1815|| ;  26763?-;  1105^;  1452f|.   74. 


3818i|;  1577^1;  20733^.    75.  2596/^;  34301  f  ;  5055j^ 


-5  * 


76.  518f f  ;  685J I ;  lOlO^^g ;  417^1 ;  548if •  77.  930|^  ; 
1230;  1812i|;  748|| ;  984.  78.  1810L|;  2-392 ^V  ^  3525 j-\  ;  1456 /^  ; 
1913f  f.  79.  2317if  ;  30622^8  ;  4512i| ;  1864  J^  ;  2449ff.  80.  12,698/^ ; 
16,77911;  24,728;  10,213|| ;  13,4232|.    81.  7289f  f  ;  9633;  14,196; 


83.  21,019|l;  27,776if ;  40,933ii;  16,907if;  22,2213^  84.  541. 
85.  723.     86.  2767^-^f.     87.  726^\o^o^. 

Page  131.  — 1.  $61.     2.  23  yd.     3.  248  weeks. 

Page  132. —4.  $130;  $650.  5.  144  doz.  6.  $53.35.  7.128 
trees.  8.  868  times.  9.  301  acres.  10.  2240  lb.  11.  150  da.;  2if  da. 
12.  72  hr.;  3  da.  13.  1569.  14.  2311,  multiplicand.  I5"  467. 
16.  92j-\lb.;  sister,  83if  lb.  17.  $51,614.  18.  $3080.  19.  97 
schooners.      20.  23f  gal.      21.  259  families. 

Page  133.  —  22.  1000.  23.  43  cars.  24.  234  da.  25.  5130  shares. 
26.  5|.Tfgmi.  27.  6272.  28.  $224.  29.  352  times.  30.  27  posts ; 
104  rails.  31.  488  acres.  32.  A,  $208;  B,  $224.  33.  3535;  3207. 
34.  779  gal. 

Page  134.  — 35.  $4.64.  36.  Elsie.  $2.65;  Lottie,  $3.05.  37.  14 
da.  38.  $584.  39.  629  mi.  40.  $  158.  41.  $  1816.  42.  $1149.12. 
43.  472  bii.  44.  12  horses ;  41  sheep.  '  45.  25  times.  46.  16  da. 
47.  Gained  $  161.      48.  50  gross. 

Page  135.  — 49.  8  75.60.  60.  5  yr.  61.  $98.79.  62.  lO^jf  vr. 
53.  Gained  .S40.  54.  938  mi.  55.  98  mi.  56.  A.  $46  ;  B,'  $  53  ; 
C.  $99.  57.  Howard,  35  tishes ;  Jamie,  23  fishes;  Roy,  19  fishes. 
58.  $198.     59.  $1410.     60.  $396. 


236  ELEMENTS   OF   ARlTHMETICo 

Page  136.  —61.  -$4725.  62.  B,  442  acres  ;  C,  1284  acres  ;  D,  1511 
acres.  63.76  1b.  64.  8  yr.  65.  .$55.  66.406,  67.22,962. 
68.  42,90a 

Page  137.  —  1.  55.      2.  30.      3.  10.      4.  2.      5.  8.      6.  5.       7.  2L 

8.  5.  9.  36.  10.  4.  11.  16.  12.  36.  13.  58.  14.  10.  15.  117. 
16.  40.     17.  112.     18.  56.     19.  109.     20.  60.     21.  157.     22.  7. 

Page  141.  —2.  2,  2,  2,  3,  7,  3.  2,  2,  2,  2,  2,  7.  4.  2,  2,  2,  2,  3,  3, 
5.  2,  3,  3,  11.  6.  2,  2,  2,  3,  3,  3.  7.  2,  2,  11,  11.  8.  2,  2,  2,  2,  2,  2, 
3,  3.  9.  2,  2,  2,  2,  3,  3,  3.  10.  2,  2,  2,  2,  2,  3,  7.  11.  2,  2,  3,  3,  11. 
12.  5,  5,  5,  5.  13.  2,  2,  2,  2,  3,  19.  14.  2,  2,  2,  2,  2,  2,  13  15.  2,  3, 
3,  5,  11.     16.  2,  2,  2,  2,  3,  3,  7.      17.  3,  5,  20.      18.  2,  3,  37.     19.  2,  2, 

2,  2,  2,  3,  7.  20.  5,  5,  31.  21.  17,  17.  22.  2,  2,  2,  2,  2,  2,  2,  2,  5. 
33.  3,  5,  5,  5,  7.  24.  2,  2,  3,  3,  89.  25.  3,  5,  7,  11.  26.  2,  2,  5,  5, 
5,  5.     27.  3,  3,  3,  5,  11.     28.  2,  2,  2,  2,  3,  3,  3,  3.      29.  2,  2,  2,  2,  2,  3, 

3,  7.     30.  5,  5,  5,  23.     31.  2,  5,  17,  17. 
Page  142. —3.  2^.     4.   1||. 

Page  143.  — 5.  4.  6.16.  7.3.  8.  27|.  9.40.  10.  18|.  11.45. 
12.  6.  13.  4.  14.  60.  15.  40.  16.  3J5.  17.  16 i.  18.  6if.  19.  8. 
20.  2[.     21.  1\.     22.  U.     23.  6f.     24.  2.     25.  1.     26.  15|  yd. "  27.  §860. 

28.  .$3. 12  V.     29.  .$3.12. 

Pagel44.  — 30.  48^.       31.  IJf.       32.  $48.       33.  53i  yd.       34.7. 
35.  39|.     36.  2||.     37.  161/^.     38.  If.     39.  4]-V.     40.  13|. 
Page  151.-2.  tf     3.  -V-.    4.  %«.     5.  -W-    G- H-     ?•  ¥•     »•  H^. 

9.  -w.    10.  W.    11-  ¥/-    12-  ¥/•    13.  -y#.    14.  W--    15-  W. 

16.  \V.     17.  iop.     18.  i||i.     19.  ^tyK    20.  ^1^2-.    21.  ^^K 
Page  152.  — 2.  17|.     3"  42f.    4.  49|.     5.  46|.    6.  52j8g.     7.  22/3. 

8.  16.l;'.        9.  12||.        10.  22i«.  11.  20t|.        12.   Uj?,.        13.  19j?y. 

14.  21||-       15.  3f|.       16.  10||.  17.  15||.       18.  24ff.       19.  49/^. 

20.  56^8.      21.  93i|. 

Pagel54.— 2. '1.     3.  f.     4.  i.  5.  }.     6.  f.     7.  i     8.  f.     9.  |. 

10.  /..      11.  2.      12.  3.      13.  ii_^  14.  11.      15.  _.!_.      16.  ^_.      17,  |. 

18.   J.     19.  I.'  20.  f.    21.  2V    22.  |.    23.  ji.    24.  f.    25.  ^V 

Paip-p   15 «5    Q       H         9         .5  ,^     1  •'■'        4         4  4     30        9         8  K      16  6 

X-age   J.OO. ,i.    j.T,    12,    J2«         «*•    18'    18 »   T8-         *•   ¥2?    T2»    T2-        *'•    3  65  3^» 

_5_        R15        8  6  71. 5        9        24         615       1218         024        6        13         lA  8 

36'      "•     20?    20'    20'        ••    ■3O'    30'    30'       **•     2l'    24'    2-f      ^-     42'    i2'    12'       •*•"•  ^?' 

6         3         2  11       1.5      12        6         4  10        6        2014        5  103320        6  66 

2¥'    2¥'    2T-       ■•■•*••     60'    60'    60'    60-       ■*''^-     ¥8'    ?8'    48'    ?F '         ■'■^-    ¥¥'   ¥¥'    ¥T'  TT* 

Pop'p    156    IR       4.5       3.5       27  Ifi         70  96  63  I7       36       2.5  14 

i-age  X3D.       10.   go,  go,  go".  lb.  jj^,  j^^y  tT2-  ^*-   eo'  eo'  eff- 

1Q      _4  8_^      __8  1  16  IQ        7  5  7  2  .5  6  Ofk       14      12      35  01        160       J7  5 

■*■**•     10,S'    10.'<'    108°  •*-^'    T2O'    T20'    T20*  *"•     ¥2'    ?2'    ¥2'  *•'••     2l)0'    20(J» 

5  6  90       100       176        3  5  OO       7  2      2  0       14  OA         8  0  3  5        2  16  OK       192 

20-0'        '*'*•     220'    220"'    220-       *"•     60'    60'   "50*        **•     T80'    T8^'    ISO'         ***•    T205 

8  8        5  2  5  OR       2  5  6       2  1       9  2  Of      4  6  5      4  2      6  5  OQ        10  2  0       118  8       10  0 

r20'   T20''  **"•    ~5G"'    56'    56-  «••    'Y'5'1    7o'    75'  ^^'     T80  '     Tf 0  '   TSO* 

29.  -Mt^-S   ^l^-   -'^4-*. 

Page  158.-2!  If.      3.  lit.  4.11^.       5.  1^-      6.   If  J.      7.  ^f. 

8.   m     9.  li^.     10.  l3fo\.     11.  10^9.     12.  162,1.     13.  16|^     14.  12|. 

15.23U.  16.  21fii.  17.  172Vo-  18.  |85,V.  19.  28' J  bu.  20.  698rVo 
Ib.    21.  125xVft.    22. 


ANSWERS.  237 

Page  160.-3.   i.    4.  3%.    5.  j\.     6.  J,.     7.  f^.     8,   H    9.   f|. 

Page  161.  - 10.  3\V.  11.  tI 0.  12-  tVo-  13-  6/^.  14.  12|i. 
15.  4,\.  16.  18^3,.  17.  43^.  18.  5|f.  19.  5^.  20.  5|t.  21.  3i|§. 
22.  10 /f..  23.  23-,\.  24.  71||lb.  25.  83 j^V  acres.  26.  6^Jft. 
27.  30 5L  lb.     28.  $30 .14.     2Q.  ^}^.     30.  2^  |.     31.  34|f  yd.    32.  ^V 

Page  162. -33,  If  34.  2if.  35.  ff  36.  y^^fo-  37.  ilJ.  38.  |i 
39.  t\.  40.  9it|.  41.  43^,.  42.  4,V,.  43.  TfJ.  44.  ^U- 
45.  41^.     46.   lOfif. 

Page  163.  — 2.  If.  3.  fa.  4.  2*.  5.  3J3.  6.  8.1.  7.  10. 
8.  lOi.  9.  12|.  10.  lOf.  11.  12i|.  12.  2Q\.  "'l3.  782."  14.  1583, 
15.  259f.         16.  130|. 

Page  164.  — 17.  71f^.  18.  247^  mi.  19.  $13.33|.  20.  $  lOOi. 
21.165  ft.        22.  16|  cords.        23.'' 175  hr.        24.  #67|. 

Page  165.  — 2,  6a.      3.  15.      4.  24/^.      5.  21.      6.  12^.      7.  20|^. 

.      13.  19i.      14.  22^. 


15.  85„        16.  2252 .         17.  241|.         18.  21|.         19.  528. 

Page  166.  — 20.  45^.  21.  100  acres.  22.  $249f.  23.  $5.61. 
24.  452f  mi.        25.  $5.03.K        26.  $148^. 

Page  168.- 2.  f.  3.  f.  4.  |.  5.  |f.  6.  [S.  7.  |.  8.  ^ 
9.  ^^0-      10.  If.      11.  j\%.      12.  iff.       13.  iff.       14.  3.       15.  6ff. 

16.  51ft.     17.  6|f.     18.  if.     19.  ItV^-     20.  4|f.     21.  26,%.     22.  J. 

23.  2^3°        24.  If.        25.  if.        26.  14/^  ft. ;  57f  ft.        27.  $  12j\. 

28.  |58J.      29.  I684  acres.      30.  $145i.      31.  $1.81}.       32.  Lining 
$2j-V;  Dress  $22f. 

Page  171.-3.  tV  4.  /^o  5./^.  G.  ^\.  7.  .V  g.  .VV 
9.  ,\.     10.   3V5.     11.  i^V     12.  life.     13.  3I2.     14.  tMt-     16=  ^h 

17.  3|f.     18.  3|f.     19.  3if.     20.  5/7.     21.  5^.    22.  17||.     23.  19if. 

24.  22ii.       25.  $8^\.       26.  4i  da.       27.  $x%perlb.       28.  $791ii. 

29.  $1|.         30.  II  da. 

Page  172.  —  31.  23 |f  mi.  per  hr.         32.  12|^  sq.  rd. 

Pagel73.  —  2.  30„  3.  36|.  4.  423.  5,  43^,  e.  54.  7.  62|. 
8.  42f.  9.  29|.  10.  791.  11.  621.  12.  72j^V  13.  76ii.  14.  47J. 
15.  36ii.  16.  45if.  17.  129.21.  18.  13931.  19.  25754.  20.  898f. 
21.  1486if.  22.  614.  23.  "91.  24.  4^^.  25.  17|!-.  26.  43^. 
27.  7||.  29.  4|f  30.  I220.  31.  7^.  32.  12/,^.  33.  9||. 
34.  20|f.  35.  522^.  36.  2J-V-*'  37.  53%.  '  38.  31^1  yd.  39.  23^  rd. 
40.  21-2«v  bbl.        41.  S6j\  bbl.        42.  3  cases. 

Page  175.-3.  Iff.  4.  l^V  5.  2^.  6.  3i.  7.  I3V 
8.  lif.  9.  lOf.  10.  2^\.  11.  3r\9^.  12.  1||.  13.  6i|. 
14.    Uff.      15.   5i|.      16.   6hl      17.   40^^.       18.   30^-°-.       19.  4r\%. 


6.3 


Page  176.— 21.    12f  times.  22.    7t3j- bu.  23.    711  bbl. 

24.  4j\  baskets.      25.  8  bbl.      26.  7|i  lb.     27.  10  books.     28.  7y^o  doz. 
29.  6  plows.    30.  5if|  acres.    31,  $4056ff.    32.  5|  bu.    33.  21  ^V?  bu. 


238  ELEMENTS  OF  ARITHMETIC, 


143 


Page  177.- 35.    .2^V      36.  J  V      37.^1^.      38.  .VVs-      39.  3/^^, 

40.  3^.       41.  n.       42.  3f^.      43.  if-      44.  ^0-      45.  jh-      46.  A\- 
2.  22.       3.   IJ3.       4.   Hi        5.  H.       6.  6|.        7o  llf.        8.  2||. 

9.    11.        10.    Iff.        11.    l^^V         12.    If.         13.   1\. 

Page  179.-2.  tV  3.  .V  4.3V  5.3V  6.  .f «.  7«  ^V 
8-  2V0.  9.  3¥t.  10.  /..  11.  2V8.  12.  t\.  13"  I-  14.  U- 
15.  33_.     16.30.     17.  tV^-.     18.  t#t.     19.  -r\V    20.^1-    21-  $1.80. 

Page  181.  — 2.  64.  3.  100.  4.  170,  5.  100.  6.  91.  7.  90. 
a.  325.       9.  400.       10.  600.      11.  850.      12.  380.      13.  510,      14.  5, 

15.  .^''o.         16.    ItV-        17.    Hf.         18.    h\'        19-   ^h 

Page  182.  — 20.  $120.       21.  100  ft.       22.  425  mi.       23.  150  ft. 
24.    192  plants.        25.   55  mi. 
Page  184.-1.  431,,     2.  63VV     3,  16|H.     4.  78f|i. 

6.  42|.        7.    765.        8.   34^1' 

Page  185. —9.  $84?.  10.  40  bu.  11.  $25jV.  12.  211f  bu. 
13.  $40831.  14.  .|6.  15.  i'..  16.  72  yr.  17,  A,  i  of  work  ; 
B,  }  of  work  ;  both,  ts  of  work.  "    18.   42  da. ;  J-  of  42  da.  =  3fV  da. 

Page  186.  — 19.  .$150,000.  20.  1,?  da.  21.  4f  da.  22.  24  da. 
23.  $620;  $465.  24.  2|  da.  25.  $90,000.  26.' 14.i^.  27.  2||. 
28.  1.  29.  8fV2.  30.  $312.50.  31.  $9000.  32.  $26,666f. 
33.    719  girls;  320  boys. 

Pagel87.  —  34.    $25,771v.V  35.    lyW/?.  36.   20//^  mi. 

37.  3|in.  ;  19||  in.  38.    $610.  39.    $250,000.  40.    ^Vfg. 

41.  Increased  yV  42.  600  1b.  43.  $  |  per  lb.  44.  $34,600. 
45.   37ida. 

Pagel92.  — 2.  f.      3.^.      4.  ^V.       5.  |.       6.1.       7o  |.       8.  f. 

9.  f.      10.  ,-^^.      11.  j\.      12.   «|o.       13.   fo-      14.  sh'      15'  rb- 

16.  ,%%.        17,  sV 

Page  193.— 2.  .3.        3.  .625.        4.   .1875.       5.   .25.       6.   .106  +  . 

7.  .15.  8.  .025.         9.   .0606  +  .         10.   .316+.         11.  ,7105263+. 

12.  .45  +  .      13.  .108+.      14.  .428751  +  .       15.   .36585  +  ,       16.  .0064. 

17.  3.5.  18.  1.857142  +  .  19.  8.66  +  .  20.  3.166  +  .  21.  6.-35,. 
22.    .571.         23.   .15625.        24.  .0375.        25.  .006. 

2.  84.331.         3.  68.454.         4.  8.301.         5.  53.875. 

Pagel94.  — 6.  33.6635.        7.  64.435.         8.  5.1195.        9.  91.7472. 

10.  430.435.  11.  8.615.  12.  49.1964.  13.  35.83262.  14.  67.8399. 
15.  80.81.  16.  1.2119.  17.  .9962493.  18.  1.938.  19.  .3706883. 
20.  133  miles.         21.  $38,741,875. 

Page  195.  — 2.  11.33.  3.10.095.  4.27.995.  5.25.46.  6.12.012. 
7.  3<).213.     8.  25.2963.     9.  5.4798.     10.  4.08.     11.  8.81.     12.  5.2905. 

13.  4.1125.       14.  .585.       15.  .161739.       16.  26.629!).       17.   12.81965. 

18.  76.5466.     19.  6.3066.     20.  4.75  bbl.      21.  $25.75.     22.   $2624.13. 
Page  196.  — 2.  .1875.     3.  .816.     4    38.76.      5.   1.545.     6.  .04371. 

7.  .34125.         8.     3285.         9.   1.19075.         10.  3.971. 


ANSWERS.  239 

Page  197.  — 11.  .013394.  12.4.02732.  13.99.8050.  14.1.3509, 
15.  17.92431.  16.   .0160655.  17.  .00103644.  18.  .001252394. 

19.  .0082622.      20.  .595875.      21.   53.9852.      22.  .051255.      24.  276.5. 

25.  496.75.  26.  279.5.  27.  893.5.  28.  687.-35.  29.  376.8.  30.  683.5. 
31.  763.4.  32.  899.2;  8992;  89920;  899200.  33.  6830.25;  68302.5; 
6&3025.  34.  .765625  Sum  greater.  35.23.7375.  36.3.03125. 
37.513.60.  38.  $30.50.         39.  391.865  acres.        40.  400.2  mi. 

Pagel98.  — 2.  17.2811.  3.   11759.6.        4.  98.765.  5.  S26.06. 

6.  $129.62.  7.  $40,598.  9.  $5,875.  10.  $7.35.  11.  $7,455. 
12.  98.0625.  13.  58.545.  14.  138.668.  15.  L4405.  16.  9.54, 
17.  .769.         18.  $655.60.  19.  $4408.375. 

Page200.  — 2.  21.7.      3.  3.9487  +  .     4.  3420.      5.  3.84.      6.  8710, 

7.  67.2,     8.  .47.     9.  3.4.     10.   .036.     11.  51.2.      12.  376.      13.  .0036. 

14.  .0131.     15.  1.461.     16.  .1671044-,     17.  .213.      18.  5.5.      19.  .036. 

20.  2500.     21.  .0072.     22.   .315.     23.  2,000.000.     24.   51.6.     25.  2.34. 

26.  100.2.  27.  64.2.  28.  6275.  29.  .0490  +  .  30.  32,820. 
31.   1,000,000,900. 

Page201.  —  33.  3.925.         34.  2.645.         35.   .3695.         36.  .04825, 

37.  .03802.     38.  .004285.     39.  26.7  yd.    40.  14.75  bbl.     41.  265.8  cords. 

Page  202. —2.  $21.82.      3.  $1564.50.      4.  $10.20.      5.   $31,405. 

Page  203. —1.   $30.  2.   $785.  3.  $58.6625.  4.  $84.50, 

5.  7000  ft.     6.  $61.-394.     7.  $4,875.     8.  8.368+  yr,     9.   1820  letters. 

10.  .36  pairs.     11.  $54.25.     12.  $121.25. 

Page204.  —  13.  13.5  hr.  14.  $17.07.  15.  $105.25.  16.  20^. 
17.  $338.        18.  $1544.70.        19.  $.023f  gain,  $  59.00.       20.  $2360. 

21.  1155.95,  tons  sold  ;  159.05,  tons  melted.         22.   12  hr. 
Page215.  —  2.   176  in.         3.  451  in.         4.  574  in.         5.  1007  in. 

6.  220  d.         7.  436  gi.         8.  494  gi.         9.  607  gi.         10.   15,315  sec. 

11.  .36,738  sec.  12.  66,035  sec.  13.  6728  oz.  14.  14,012  oz. 

15.  73,292  oz.  16.  139,769  oz.  17.  26,3(>2  gr.  18.  48.248  gr. 
19.  31,-569  gr,  20.  251  pt.  21.  -367  pt.  22.  409  pt.  23.  930  in. 
24.  1082  in.  25.  1654  in.  26.  4482  sq.  in.  27.  7732  sq.  in. 
28.  13,012^  sq.  yd.  29.  -39,480]  sq.  yd.  30.  110,677  cu.  in. 
31.  197,706  cu.  in.  32.  18,680  gr.  33.  27,220  gr.  34.  1698  sheets. 
35.  2122  sheets.  36.  20,400  gr.  37.  33,660  gr.  38.  555,900  ct. 
39.  575  pt.  40.  1758  in.  41.  1980  pt.  42.  28,438  sq.  yd. 
43.  463,482  in.  45.  37  lb.  8  oz.  46.  3  yd.  1  ft.  3|  in.  47.  5  da. 
10  hr.  40  min.  48.  1  ft.  3i  in.  49.  2  pk.  3.2  qt.  50.  9  cwt. 
61.  10  cu.  ft.  1382.4  cu.  in.  '  52.  266  rd.  3  yd.  2  ft.  53.  68  sq.  rd. 
17  sq.  yd.  2  sq.  ft.  82^  sq.  in.  54.  5  da.  2  hr.  38  min.  24  sec. 
55.  1  qt.  2  gi.     56.   1  ft.  5.1  in. 

Page  216.  —3.  131  gal.  2  qt.  1  pt.  1  gi.  4.  398  s^al.  1  pt.  5.  178 
gal.  3  qt.  1  pt.  6.    17  rd.  4.^  yd.  11  in.  7.  ^2  rd.  2  yd.  8  in. 

8.  8  rd.  1  ft.  9  in. 

Page  217.  —9.  55  bu.  2  pk.  6  qt.  10.  73  bu.  5  qt.  1  pt.  11.  61  bu. 
2  pk.  5  qt.  1  pt.  12.  3  cwt.  22  lb.  12  oz.  13.  5  cwt.  7  lb.  15  oz. 
14.  2  T.  19  cwt.  83  lb.  15.  1  T.  17  cwt.  95  lb.  16.  25  wk.  3  da. 
13  hr.  17.  2  hr.  16  min.  8  sec.  18.  6  da.  20  hr.  54  min.  19.  3  sq.  yd. 
55  sq.  in.   20.  4  sq.  yd.  5  sq.  ft.  33  sq.  in.   21.  3  R.  18  qr.  22  sheets. 

22.  4  R.  16  qr.  11  sheets.   23.  4  cu.  ft.  952  cu.  in.   24.  1  lb.  2  oz. 


240 


ELEMENTS   OF   ARITHMETIC. 


1  pwt.  10  gr.    25.  1  lb.  5  oz.  9  pwt.  10  gr.    26.  1  lb.  7  oz.  18  gr. 
28.  i\yd.  29.  Jspk.  30.  l/.ybu.  31.  p  gal.  32.  |-i  bu.  33.  ff  rd. 

Page  218.  —  2.  36  gal.  1  ql  3.  38  bu.  2  pk.  4.  22  da.  13  hi. 
30  mill.   5.  169  lb.  14  oz.   6.  35  lb.  8  oz.  17  pwt.  21  gr.   7.  70  yd. 

2  ft.  9  in.  8.  84  gal.  1  qt.  1  pt.  9.  79  A.  153  sq.  rd.  10.  41  T.  31  lb. 
11.  23  yd.  1  ft.  lOi  in.  2.  5  bu.  1  pk.  4  qt. 

Page  219.  —  3.  6  gal.  2  qt.  1  pt.  3  gi.  4.  2  da.  22  hr.  10  min. 
5.  22  lb.  10  oz.  3  pwt.  6.  10  yd.  2 It.  9  in.  7.  3  rd.  4i-  yd.  2  ft. 
8.  5  lb.  9  oz.  6  dr.  9.  10  A.  68  sq.  rd.  10.  25  sq.  yd.  1  sq.  ft.  95  sq.  in. 
11.  14  T.  17  cwt.  39  lb.  11  oz.  13.  51  yr.  5  mo.  5  da.  14.  25  yr. 
6  mo.  18  da.  15.  52  yr.  2  mo.  29  da.  16.  10  yr.  10  mo.  17  da. 

Page  220.  —  2.  34  bu.  1  pk.  3  qt.  3.  34  gal.  1  pt.  4.  44  lb.  4  oz. 
2  pwt.  16  gr.  5.  54  lb.  1  oz.  2  sc.  10  gr.  6.  16  hr.  42  min.  55  sec. 
7.  18  T.  3  cwt.  91  lb.  8  oz.  8.  24  rd.  2  yd.  1  ft.  6  in.  9.  75  cu.  yd. 
12  cu.  ft.  1506  cu.  in.  10.  41  sq.  yd.  2  sq.  ft.  81  sq.  in.  11.  21  da. 
9  hr.  1  min.  40  sec.  12.  103  rd.  1  ft.  10  in.  13.  48  R.  18  qr. 
14.  44  bbl.  13  gal.  1  pt. 

2.  3  gal.  2  qt.  1  pt.  li  gi.  3.  3  bu.  3  pk.  7  qt.  f  pt.  4.  1  yd.  2  ft. 
11}  in.  5.  4  cwt.  73  lb.  llf  oz.  6.  4  lb.  8  oz.  19  pwt.  10|  gr. 
7.  2  lb.  9  oz.  7  dr.  2  sc.  2^  gr.  8.  3  hr.  11  min.  46  sec.  9.  5  sq.  yd. 
4  sq.  ft.  Hi  sq.  in.    lO;"  3  rd.  2  ft.  llj  in. 

Page  222.  —  2.  391  sq.  ft.  3.  192  sq.  ft. ;  211  gq.  yd.  4.  48  sq.  yd. 


6.  30  A.   7.  $17,578+.   8 


12.  15  A.  corn 
3.  792  cu.  in. 

7.  888  cu,  ft. 
11.  3  cd.   12 


7i 
4. 
8. 

8  cd. 


5,  19,200  sq.  rd.  ;  120  A. 
9.  $25.50.   10.  240  A.   11.  $60. 
Page  224.-2.  420  cu.  ft. 

5.  2100  cu.  ft.  6.  680  cu.  ft. 
9.  986f  cu.  ft  10.  700  cu.  ft. 
14.  $396. 

Page  225.  — 15.  $78,222.    16.  $36.3636  +  . 
1.  24  ft.   2.  15  ft.   3.  13L  ft.   4.  23i  ft.   5.  12*  ft. 
7.  $3.00.   8.  1440ft.   9.  $25.92.   10.  $16,426+": 
Page  227.-2.  $25.4808.  3.  $85,224.  4.  $97,275. 

6.  $64,832.    7.  $174.69. 
Page  228.-9.  $43,968.  10.  $66.8556.  11.  $93,907. 


$27.14. 

A.  potatoes. 

400  cu.  yd. 

1260  cu.  ft. 

13.  72  cd. 


5.  26 1  ft. 

$  25.38. 

$61,376. 


12 


13. 

Amt. 
Amt. 
Amt. 
Amt. 
Amt. 


180.742 

$407,521. 

$239,458. 

$850,888. 

$251,732. 

$  527.838. 

Page  229.  —1. 

4.  9  bu.  2  pk.  5  qt. 

8.  !«-.  9.  5831  bu. 

Page  230.— 13 


14.  $40 

18.  $85,865 
20.  $94,195 
22.  $70,774 
24.  $25,493 
26.  $94.44; 
!  250.    2. 
5.  1685:fV 
10.  4i  da. 
$36, 


15.    $125,147. 

Amt.  $501,345. 
Amt.  $  517.545. 
Amt.   $539,134. 
Amt.  $350.8339. 
\mt.  $391.89. 
Answers  differ.        3. 
revolutions.     6.  95^\ 

11.  $26250.     12.  40/' 
14.  105  A.  145  sq.  rd. 


17. 
19. 
21. 
23. 
25. 


$171.97. 
$47.2815; 
$22,098; 
$130,728; 
$32,262; 
$91,508; 


4  yd.  1  ft.  10  in. 
7.  10  children. 
;  25</. 

15.  75.*  A 


16.  $  7.87  amount  of  bill.  17.  5  da.  5  h.  33  min.  20  sec.  18.  453^  mi. 
19.2160  ft.  20.  $37,453  +  .  21.  $  161.906  +  .  22.  5  lb.  2  oz.  23.48 
suits.     24.  $40.     25.  42  yd.;  $29.40.     26.  64  rd.  long. 

Page  231.  — 27.  $582.40.     28.  210  ;  280.     29.  $62500.     30.  $633; 
$844.     31.   $413.     32.  $55.20.     33.  2880  pills.     34.  $29.      35.  $  5.04. 
36.  $50;  $60.     37.  $39.20.     38._f|.     39.  7|f  mi.    40.   13.4. 
^\  S  B  A  rt  y 

UNIVERSITY 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


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-    JUL  16  1917 

0CT21M8 
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jin->  O-^  ?«■?« 


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%  3  r 


